Theorem poimirlem28

Description: Lemma for poimir , a variant of Sperner's lemma. (Contributed by Brendan Leahy, 21-Aug-2020)

	Ref	Expression
Hypotheses	poimir.0	$$⊢\phi \to N\in \mathbb{N}$$
	poimirlem28.1	$$⊢p=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\to B=C$$
	poimirlem28.2	$$⊢\left(\phi \wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\right)\to B\in \left(0\dots N\right)$$
	poimirlem28.3	$$⊢\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\to B<n$$
	poimirlem28.4	$$⊢\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=K\right)\right)\to B\ne n-1$$
	poimirlem28.5	$$⊢\phi \to K\in \mathbb{N}$$
Assertion	poimirlem28	$$⊢\phi \to \exists s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C$$

Proof

Step	Hyp	Ref	Expression
1		poimir.0	$$⊢\phi \to N\in \mathbb{N}$$
2		poimirlem28.1	$$⊢p=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\to B=C$$
3		poimirlem28.2	$$⊢\left(\phi \wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\right)\to B\in \left(0\dots N\right)$$
4		poimirlem28.3	$$⊢\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\to B<n$$
5		poimirlem28.4	$$⊢\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=K\right)\right)\to B\ne n-1$$
6		poimirlem28.5	$$⊢\phi \to K\in \mathbb{N}$$
7	1	nnnn0d	$$⊢\phi \to N\in {\mathbb{N}}_0$$
8	1	nnred	$$⊢\phi \to N\in ℝ$$
9	8	leidd	$$⊢\phi \to N\le N$$
10	7 7 9	3jca	$$⊢\phi \to \left(N\in {\mathbb{N}}_0\wedge N\in {\mathbb{N}}_0\wedge N\le N\right)$$
11		oveq2	$$⊢k=0\to \left(1\dots k\right)=\left(1\dots 0\right)$$
12		fz10	$$⊢\left(1\dots 0\right)=\varnothing$$
13	11 12	eqtrdi	$$⊢k=0\to \left(1\dots k\right)=\varnothing$$
14	13	oveq2d	$$⊢k=0\to {\left(0..^K\right)}^{\left(1\dots k\right)}={\left(0..^K\right)}^{\varnothing }$$
15		fzofi	$$⊢\left(0..^K\right)\in \mathrm{Fin}$$
16		map0e	$$⊢\left(0..^K\right)\in \mathrm{Fin}\to {\left(0..^K\right)}^{\varnothing }=1_{𝑜}$$
17	15 16	ax-mp	$$⊢{\left(0..^K\right)}^{\varnothing }=1_{𝑜}$$
18		df1o2	$$⊢1_{𝑜}=\left\{\varnothing \right\}$$
19	17 18	eqtri	$$⊢{\left(0..^K\right)}^{\varnothing }=\left\{\varnothing \right\}$$
20	14 19	eqtrdi	$$⊢k=0\to {\left(0..^K\right)}^{\left(1\dots k\right)}=\left\{\varnothing \right\}$$
21		eqidd	$$⊢k=0\to f=f$$
22	21 13 13	f1oeq123d	$$⊢k=0\to \left(f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\leftrightarrow f:\varnothing \underset{1-1\; onto}{⟶}\varnothing \right)$$
23		eqid	$$⊢\varnothing =\varnothing$$
24		f1o00	$$⊢f:\varnothing \underset{1-1\; onto}{⟶}\varnothing \leftrightarrow \left(f=\varnothing \wedge \varnothing =\varnothing \right)$$
25	23 24	mpbiran2	$$⊢f:\varnothing \underset{1-1\; onto}{⟶}\varnothing \leftrightarrow f=\varnothing$$
26	22 25	bitrdi	$$⊢k=0\to \left(f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\leftrightarrow f=\varnothing \right)$$
27	26	abbidv	$$⊢k=0\to \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left\{f|f=\varnothing \right\}$$
28		df-sn	$$⊢\left\{\varnothing \right\}=\left\{f|f=\varnothing \right\}$$
29	27 28	eqtr4di	$$⊢k=0\to \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left\{\varnothing \right\}$$
30	20 29	xpeq12d	$$⊢k=0\to \left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left\{\varnothing \right\}\times \left\{\varnothing \right\}$$
31		0ex	$$⊢\varnothing \in \mathrm{V}$$
32	31 31	xpsn	$$⊢\left\{\varnothing \right\}\times \left\{\varnothing \right\}=\left\{⟨\varnothing ,\varnothing ⟩\right\}$$
33	30 32	eqtr2di	$$⊢k=0\to \left\{⟨\varnothing ,\varnothing ⟩\right\}=\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}$$
34		elsni	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to s=⟨\varnothing ,\varnothing ⟩$$
35	31 31	op1std	$$⊢s=⟨\varnothing ,\varnothing ⟩\to 1^{st}\left(s\right)=\varnothing$$
36	34 35	syl	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to 1^{st}\left(s\right)=\varnothing$$
37	36	oveq1d	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to 1^{st}\left(s\right){+}_f\varnothing =\varnothing {+}_f\varnothing$$
38		f0	$$⊢\varnothing :\varnothing ⟶\varnothing$$
39		ffn	$$⊢\varnothing :\varnothing ⟶\varnothing \to \varnothing Fn\varnothing$$
40	38 39	mp1i	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to \varnothing Fn\varnothing$$
41	31	a1i	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to \varnothing \in \mathrm{V}$$
42		inidm	$$⊢\varnothing \cap \varnothing =\varnothing$$
43		0fv	$$⊢\varnothing \left(n\right)=\varnothing$$
44	43	a1i	$$⊢\left(s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\wedge n\in \varnothing \right)\to \varnothing \left(n\right)=\varnothing$$
45	40 40 41 41 42 44 44	offval	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to \varnothing {+}_f\varnothing =\left(n\in \varnothing ⟼\varnothing +\varnothing \right)$$
46		mpt0	$$⊢\left(n\in \varnothing ⟼\varnothing +\varnothing \right)=\varnothing$$
47	45 46	eqtrdi	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to \varnothing {+}_f\varnothing =\varnothing$$
48	37 47	eqtrd	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to 1^{st}\left(s\right){+}_f\varnothing =\varnothing$$
49	48	uneq1d	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to \left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)=\varnothing \cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)$$
50		uncom	$$⊢\varnothing \cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)=\left(\left(1\dots N\right)\times \left\{0\right\}\right)\cup \varnothing$$
51		un0	$$⊢\left(\left(1\dots N\right)\times \left\{0\right\}\right)\cup \varnothing =\left(1\dots N\right)\times \left\{0\right\}$$
52	50 51	eqtri	$$⊢\varnothing \cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)=\left(1\dots N\right)\times \left\{0\right\}$$
53	49 52	eqtr2di	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to \left(1\dots N\right)\times \left\{0\right\}=\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)$$
54	53	csbeq1d	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to ⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
55	54	eqeq2d	$$⊢s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\to \left(0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\leftrightarrow 0=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
56		oveq2	$$⊢k=0\to \left(0\dots k\right)=\left(0\dots 0\right)$$
57		0z	$$⊢0\in \mathbb{Z}$$
58		fzsn	$$⊢0\in \mathbb{Z}\to \left(0\dots 0\right)=\left\{0\right\}$$
59	57 58	ax-mp	$$⊢\left(0\dots 0\right)=\left\{0\right\}$$
60	56 59	eqtrdi	$$⊢k=0\to \left(0\dots k\right)=\left\{0\right\}$$
61		oveq2	$$⊢k=0\to \left(j+1\dots k\right)=\left(j+1\dots 0\right)$$
62	61	imaeq2d	$$⊢k=0\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]=2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]$$
63	62	xpeq1d	$$⊢k=0\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}=2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}$$
64	63	uneq2d	$$⊢k=0\to \left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)=\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)$$
65	64	oveq2d	$$⊢k=0\to 1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)$$
66		oveq1	$$⊢k=0\to k+1=0+1$$
67		0p1e1	$$⊢0+1=1$$
68	66 67	eqtrdi	$$⊢k=0\to k+1=1$$
69	68	oveq1d	$$⊢k=0\to \left(k+1\dots N\right)=\left(1\dots N\right)$$
70	69	xpeq1d	$$⊢k=0\to \left(k+1\dots N\right)\times \left\{0\right\}=\left(1\dots N\right)\times \left\{0\right\}$$
71	65 70	uneq12d	$$⊢k=0\to \left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)=\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)$$
72	71	csbeq1d	$$⊢k=0\to ⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
73	72	eqeq2d	$$⊢k=0\to \left(i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
74	60 73	rexeqbidv	$$⊢k=0\to \left(\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \exists j\in \left\{0\right\}i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
75		c0ex	$$⊢0\in \mathrm{V}$$
76		oveq2	$$⊢j=0\to \left(1\dots j\right)=\left(1\dots 0\right)$$
77	76 12	eqtrdi	$$⊢j=0\to \left(1\dots j\right)=\varnothing$$
78	77	imaeq2d	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]=2^{nd}\left(s\right)\left[\varnothing \right]$$
79		ima0	$$⊢2^{nd}\left(s\right)\left[\varnothing \right]=\varnothing$$
80	78 79	eqtrdi	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]=\varnothing$$
81	80	xpeq1d	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}=\varnothing \times \left\{1\right\}$$
82		0xp	$$⊢\varnothing \times \left\{1\right\}=\varnothing$$
83	81 82	eqtrdi	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}=\varnothing$$
84		oveq1	$$⊢j=0\to j+1=0+1$$
85	84 67	eqtrdi	$$⊢j=0\to j+1=1$$
86	85	oveq1d	$$⊢j=0\to \left(j+1\dots 0\right)=\left(1\dots 0\right)$$
87	86 12	eqtrdi	$$⊢j=0\to \left(j+1\dots 0\right)=\varnothing$$
88	87	imaeq2d	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]=2^{nd}\left(s\right)\left[\varnothing \right]$$
89	88 79	eqtrdi	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]=\varnothing$$
90	89	xpeq1d	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}=\varnothing \times \left\{0\right\}$$
91		0xp	$$⊢\varnothing \times \left\{0\right\}=\varnothing$$
92	90 91	eqtrdi	$$⊢j=0\to 2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}=\varnothing$$
93	83 92	uneq12d	$$⊢j=0\to \left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)=\varnothing \cup \varnothing$$
94		un0	$$⊢\varnothing \cup \varnothing =\varnothing$$
95	93 94	eqtrdi	$$⊢j=0\to \left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)=\varnothing$$
96	95	oveq2d	$$⊢j=0\to 1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)=1^{st}\left(s\right){+}_f\varnothing$$
97	96	uneq1d	$$⊢j=0\to \left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)=\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)$$
98	97	csbeq1d	$$⊢j=0\to ⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
99	98	eqeq2d	$$⊢j=0\to \left(i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
100	75 99	rexsn	$$⊢\exists j\in \left\{0\right\}i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots 0\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
101	74 100	bitrdi	$$⊢k=0\to \left(\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
102	60 101	raleqbidv	$$⊢k=0\to \left(\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \forall i\in \left\{0\right\}i=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
103		eqeq1	$$⊢i=0\to \left(i=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow 0=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
104	75 103	ralsn	$$⊢\forall i\in \left\{0\right\}i=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow 0=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
105	102 104	bitr2di	$$⊢k=0\to \left(0=⦋\left(\left(1^{st}\left(s\right){+}_f\varnothing \right)\cup \left(\left(1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
106	55 105	sylan9bbr	$$⊢\left(k=0\wedge s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}\right)\to \left(0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\leftrightarrow \forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
107	33 106	rabeqbidva	$$⊢k=0\to \left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}=\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}$$
108	107	eqcomd	$$⊢k=0\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}=\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}$$
109	108	fveq2d	$$⊢k=0\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|$$
110	109	breq2d	$$⊢k=0\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|\right)$$
111	110	notbid	$$⊢k=0\to \left(\neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow \neg 2\parallel \left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|\right)$$
112	111	imbi2d	$$⊢k=0\to \left(\left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\leftrightarrow \left(\phi \to \neg 2\parallel \left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|\right)\right)$$
113		oveq2	$$⊢k=m\to \left(1\dots k\right)=\left(1\dots m\right)$$
114	113	oveq2d	$$⊢k=m\to {\left(0..^K\right)}^{\left(1\dots k\right)}={\left(0..^K\right)}^{\left(1\dots m\right)}$$
115		eqidd	$$⊢k=m\to f=f$$
116	115 113 113	f1oeq123d	$$⊢k=m\to \left(f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\leftrightarrow f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right)$$
117	116	abbidv	$$⊢k=m\to \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}$$
118	114 117	xpeq12d	$$⊢k=m\to \left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}$$
119		oveq2	$$⊢k=m\to \left(0\dots k\right)=\left(0\dots m\right)$$
120		oveq2	$$⊢k=m\to \left(j+1\dots k\right)=\left(j+1\dots m\right)$$
121	120	imaeq2d	$$⊢k=m\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]=2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]$$
122	121	xpeq1d	$$⊢k=m\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}=2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}$$
123	122	uneq2d	$$⊢k=m\to \left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)=\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)$$
124	123	oveq2d	$$⊢k=m\to 1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)$$
125		oveq1	$$⊢k=m\to k+1=m+1$$
126	125	oveq1d	$$⊢k=m\to \left(k+1\dots N\right)=\left(m+1\dots N\right)$$
127	126	xpeq1d	$$⊢k=m\to \left(k+1\dots N\right)\times \left\{0\right\}=\left(m+1\dots N\right)\times \left\{0\right\}$$
128	124 127	uneq12d	$$⊢k=m\to \left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)=\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)$$
129	128	csbeq1d	$$⊢k=m\to ⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
130	129	eqeq2d	$$⊢k=m\to \left(i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
131	119 130	rexeqbidv	$$⊢k=m\to \left(\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
132	119 131	raleqbidv	$$⊢k=m\to \left(\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
133	118 132	rabeqbidv	$$⊢k=m\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}=\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}$$
134	133	fveq2d	$$⊢k=m\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|$$
135	134	breq2d	$$⊢k=m\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
136	135	notbid	$$⊢k=m\to \left(\neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
137	136	imbi2d	$$⊢k=m\to \left(\left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\leftrightarrow \left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
138		oveq2	$$⊢k=m+1\to \left(1\dots k\right)=\left(1\dots m+1\right)$$
139	138	oveq2d	$$⊢k=m+1\to {\left(0..^K\right)}^{\left(1\dots k\right)}={\left(0..^K\right)}^{\left(1\dots m+1\right)}$$
140		eqidd	$$⊢k=m+1\to f=f$$
141	140 138 138	f1oeq123d	$$⊢k=m+1\to \left(f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\leftrightarrow f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right)$$
142	141	abbidv	$$⊢k=m+1\to \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}$$
143	139 142	xpeq12d	$$⊢k=m+1\to \left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}$$
144		oveq2	$$⊢k=m+1\to \left(0\dots k\right)=\left(0\dots m+1\right)$$
145		oveq2	$$⊢k=m+1\to \left(j+1\dots k\right)=\left(j+1\dots m+1\right)$$
146	145	imaeq2d	$$⊢k=m+1\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]=2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]$$
147	146	xpeq1d	$$⊢k=m+1\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}=2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}$$
148	147	uneq2d	$$⊢k=m+1\to \left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)=\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)$$
149	148	oveq2d	$$⊢k=m+1\to 1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)$$
150		oveq1	$$⊢k=m+1\to k+1=m+1+1$$
151	150	oveq1d	$$⊢k=m+1\to \left(k+1\dots N\right)=\left(m+1+1\dots N\right)$$
152	151	xpeq1d	$$⊢k=m+1\to \left(k+1\dots N\right)\times \left\{0\right\}=\left(m+1+1\dots N\right)\times \left\{0\right\}$$
153	149 152	uneq12d	$$⊢k=m+1\to \left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)=\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)$$
154	153	csbeq1d	$$⊢k=m+1\to ⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
155	154	eqeq2d	$$⊢k=m+1\to \left(i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
156	144 155	rexeqbidv	$$⊢k=m+1\to \left(\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
157	144 156	raleqbidv	$$⊢k=m+1\to \left(\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
158	143 157	rabeqbidv	$$⊢k=m+1\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}=\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}$$
159	158	fveq2d	$$⊢k=m+1\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|$$
160	159	breq2d	$$⊢k=m+1\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
161	160	notbid	$$⊢k=m+1\to \left(\neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
162	161	imbi2d	$$⊢k=m+1\to \left(\left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\leftrightarrow \left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
163		oveq2	$$⊢k=N\to \left(1\dots k\right)=\left(1\dots N\right)$$
164	163	oveq2d	$$⊢k=N\to {\left(0..^K\right)}^{\left(1\dots k\right)}={\left(0..^K\right)}^{\left(1\dots N\right)}$$
165		eqidd	$$⊢k=N\to f=f$$
166	165 163 163	f1oeq123d	$$⊢k=N\to \left(f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\leftrightarrow f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right)$$
167	166	abbidv	$$⊢k=N\to \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}$$
168	164 167	xpeq12d	$$⊢k=N\to \left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}=\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}$$
169		oveq2	$$⊢k=N\to \left(0\dots k\right)=\left(0\dots N\right)$$
170		oveq2	$$⊢k=N\to \left(j+1\dots k\right)=\left(j+1\dots N\right)$$
171	170	imaeq2d	$$⊢k=N\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]=2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]$$
172	171	xpeq1d	$$⊢k=N\to 2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}=2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}$$
173	172	uneq2d	$$⊢k=N\to \left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)=\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)$$
174	173	oveq2d	$$⊢k=N\to 1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)$$
175		oveq1	$$⊢k=N\to k+1=N+1$$
176	175	oveq1d	$$⊢k=N\to \left(k+1\dots N\right)=\left(N+1\dots N\right)$$
177	176	xpeq1d	$$⊢k=N\to \left(k+1\dots N\right)\times \left\{0\right\}=\left(N+1\dots N\right)\times \left\{0\right\}$$
178	174 177	uneq12d	$$⊢k=N\to \left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)=\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)$$
179	178	csbeq1d	$$⊢k=N\to ⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
180	179	eqeq2d	$$⊢k=N\to \left(i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
181	169 180	rexeqbidv	$$⊢k=N\to \left(\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
182	169 181	raleqbidv	$$⊢k=N\to \left(\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
183	168 182	rabeqbidv	$$⊢k=N\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}=\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}$$
184	183	fveq2d	$$⊢k=N\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|$$
185	184	breq2d	$$⊢k=N\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
186	185	notbid	$$⊢k=N\to \left(\neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
187	186	imbi2d	$$⊢k=N\to \left(\left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots k\right)}\right)\times \left\{f|f:\left(1\dots k\right)\underset{1-1\; onto}{⟶}\left(1\dots k\right)\right\}\right)|\forall i\in \left(0\dots k\right)\exists j\in \left(0\dots k\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots k\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(k+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\leftrightarrow \left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
188		n2dvds1	$$⊢\neg 2\parallel 1$$
189		opex	$$⊢⟨\varnothing ,\varnothing ⟩\in \mathrm{V}$$
190		hashsng	$$⊢⟨\varnothing ,\varnothing ⟩\in \mathrm{V}\to \left|\left\{⟨\varnothing ,\varnothing ⟩\right\}\right|=1$$
191	189 190	ax-mp	$$⊢\left|\left\{⟨\varnothing ,\varnothing ⟩\right\}\right|=1$$
192		nnuz	$$⊢\mathbb{N}={\mathbb{Z}}_{\ge 1}$$
193	1 192	eleqtrdi	$$⊢\phi \to N\in {\mathbb{Z}}_{\ge 1}$$
194		eluzfz1	$$⊢N\in {\mathbb{Z}}_{\ge 1}\to 1\in \left(1\dots N\right)$$
195	193 194	syl	$$⊢\phi \to 1\in \left(1\dots N\right)$$
196	6	nnnn0d	$$⊢\phi \to K\in {\mathbb{N}}_0$$
197		0elfz	$$⊢K\in {\mathbb{N}}_0\to 0\in \left(0\dots K\right)$$
198		fconst6g	$$⊢0\in \left(0\dots K\right)\to \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
199	196 197 198	3syl	$$⊢\phi \to \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
200	75	fvconst2	$$⊢1\in \left(1\dots N\right)\to \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0$$
201	195 200	syl	$$⊢\phi \to \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0$$
202	195 199 201	3jca	$$⊢\phi \to \left(1\in \left(1\dots N\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)$$
203		nfv	$$⊢Ⅎp\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)\right)$$
204		nfcsb1v	$$⊢\underset{\_}{Ⅎ}p⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B$$
205	204	nfeq1	$$⊢Ⅎp⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B=0$$
206	203 205	nfim	$$⊢Ⅎp\left(\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)\right)\to ⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B=0\right)$$
207		ovex	$$⊢\left(1\dots N\right)\in \mathrm{V}$$
208		snex	$$⊢\left\{0\right\}\in \mathrm{V}$$
209	207 208	xpex	$$⊢\left(1\dots N\right)\times \left\{0\right\}\in \mathrm{V}$$
210		feq1	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to \left(p:\left(1\dots N\right)⟶\left(0\dots K\right)\leftrightarrow \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)$$
211		fveq1	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to p\left(1\right)=\left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)$$
212	211	eqeq1d	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to \left(p\left(1\right)=0\leftrightarrow \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)$$
213	210 212	3anbi23d	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to \left(\left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\leftrightarrow \left(1\in \left(1\dots N\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)\right)$$
214	213	anbi2d	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to \left(\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)\leftrightarrow \left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)\right)\right)$$
215		csbeq1a	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to B=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B$$
216	215	eqeq1d	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to \left(B=0\leftrightarrow ⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B=0\right)$$
217	214 216	imbi12d	$$⊢p=\left(1\dots N\right)\times \left\{0\right\}\to \left(\left(\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)\to B=0\right)\leftrightarrow \left(\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)\right)\to ⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B=0\right)\right)$$
218		1ex	$$⊢1\in \mathrm{V}$$
219		eleq1	$$⊢n=1\to \left(n\in \left(1\dots N\right)\leftrightarrow 1\in \left(1\dots N\right)\right)$$
220		fveqeq2	$$⊢n=1\to \left(p\left(n\right)=0\leftrightarrow p\left(1\right)=0\right)$$
221	219 220	3anbi13d	$$⊢n=1\to \left(\left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\leftrightarrow \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)$$
222	221	anbi2d	$$⊢n=1\to \left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\leftrightarrow \left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)\right)$$
223		breq2	$$⊢n=1\to \left(B<n\leftrightarrow B<1\right)$$
224	222 223	imbi12d	$$⊢n=1\to \left(\left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\to B<n\right)\leftrightarrow \left(\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)\to B<1\right)\right)$$
225	218 224 4	vtocl	$$⊢\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)\to B<1$$
226		elfznn0	$$⊢B\in \left(0\dots N\right)\to B\in {\mathbb{N}}_0$$
227		nn0lt10b	$$⊢B\in {\mathbb{N}}_0\to \left(B<1\leftrightarrow B=0\right)$$
228	3 226 227	3syl	$$⊢\left(\phi \wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\right)\to \left(B<1\leftrightarrow B=0\right)$$
229	228	3ad2antr2	$$⊢\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)\to \left(B<1\leftrightarrow B=0\right)$$
230	225 229	mpbid	$$⊢\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(1\right)=0\right)\right)\to B=0$$
231	206 209 217 230	vtoclf	$$⊢\left(\phi \wedge \left(1\in \left(1\dots N\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(\left(1\dots N\right)\times \left\{0\right\}\right)\left(1\right)=0\right)\right)\to ⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B=0$$
232	202 231	mpdan	$$⊢\phi \to ⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B=0$$
233	232	eqcomd	$$⊢\phi \to 0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B$$
234	233	ralrimivw	$$⊢\phi \to \forall s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B$$
235		rabid2	$$⊢\left\{⟨\varnothing ,\varnothing ⟩\right\}=\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\leftrightarrow \forall s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B$$
236	234 235	sylibr	$$⊢\phi \to \left\{⟨\varnothing ,\varnothing ⟩\right\}=\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}$$
237	236	fveq2d	$$⊢\phi \to \left|\left\{⟨\varnothing ,\varnothing ⟩\right\}\right|=\left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|$$
238	191 237	eqtr3id	$$⊢\phi \to 1=\left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|$$
239	238	breq2d	$$⊢\phi \to \left(2\parallel 1\leftrightarrow 2\parallel \left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|\right)$$
240	188 239	mtbii	$$⊢\phi \to \neg 2\parallel \left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|$$
241	240	a1i	$$⊢N\in {\mathbb{N}}_0\to \left(\phi \to \neg 2\parallel \left|\left\{s\in \left\{⟨\varnothing ,\varnothing ⟩\right\}|0=⦋\left(\left(1\dots N\right)\times \left\{0\right\}\right)/p⦌B\right\}\right|\right)$$
242		2z	$$⊢2\in \mathbb{Z}$$
243		fzfi	$$⊢\left(1\dots m+1\right)\in \mathrm{Fin}$$
244		mapfi	$$⊢\left(\left(0..^K\right)\in \mathrm{Fin}\wedge \left(1\dots m+1\right)\in \mathrm{Fin}\right)\to {\left(0..^K\right)}^{\left(1\dots m+1\right)}\in \mathrm{Fin}$$
245	15 243 244	mp2an	$$⊢{\left(0..^K\right)}^{\left(1\dots m+1\right)}\in \mathrm{Fin}$$
246		ovex	$$⊢\left(1\dots m+1\right)\in \mathrm{V}$$
247	246 246	mapval	$$⊢{\left(1\dots m+1\right)}^{\left(1\dots m+1\right)}=\left\{f|f:\left(1\dots m+1\right)⟶\left(1\dots m+1\right)\right\}$$
248		mapfi	$$⊢\left(\left(1\dots m+1\right)\in \mathrm{Fin}\wedge \left(1\dots m+1\right)\in \mathrm{Fin}\right)\to {\left(1\dots m+1\right)}^{\left(1\dots m+1\right)}\in \mathrm{Fin}$$
249	243 243 248	mp2an	$$⊢{\left(1\dots m+1\right)}^{\left(1\dots m+1\right)}\in \mathrm{Fin}$$
250	247 249	eqeltrri	$$⊢\left\{f|f:\left(1\dots m+1\right)⟶\left(1\dots m+1\right)\right\}\in \mathrm{Fin}$$
251		f1of	$$⊢f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\to f:\left(1\dots m+1\right)⟶\left(1\dots m+1\right)$$
252	251	ss2abi	$$⊢\left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\subseteq \left\{f|f:\left(1\dots m+1\right)⟶\left(1\dots m+1\right)\right\}$$
253		ssfi	$$⊢\left(\left\{f|f:\left(1\dots m+1\right)⟶\left(1\dots m+1\right)\right\}\in \mathrm{Fin}\wedge \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\subseteq \left\{f|f:\left(1\dots m+1\right)⟶\left(1\dots m+1\right)\right\}\right)\to \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}$$
254	250 252 253	mp2an	$$⊢\left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}$$
255		xpfi	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\in \mathrm{Fin}\wedge \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}\right)\to \left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}$$
256	245 254 255	mp2an	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}$$
257		rabfi	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\in \mathrm{Fin}$$
258		hashcl	$$⊢\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\in \mathrm{Fin}\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in {\mathbb{N}}_0$$
259	256 257 258	mp2b	$$⊢\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in {\mathbb{N}}_0$$
260	259	nn0zi	$$⊢\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}$$
261		rabfi	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\in \mathrm{Fin}$$
262		hashcl	$$⊢\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\in \mathrm{Fin}\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in {\mathbb{N}}_0$$
263	256 261 262	mp2b	$$⊢\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in {\mathbb{N}}_0$$
264	263	nn0zi	$$⊢\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in \mathbb{Z}$$
265	242 260 264	3pm3.2i	$$⊢\left(2\in \mathbb{Z}\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in \mathbb{Z}\right)$$
266		nn0p1nn	$$⊢m\in {\mathbb{N}}_0\to m+1\in \mathbb{N}$$
267	266	ad2antrl	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to m+1\in \mathbb{N}$$
268		uneq1	$$⊢q=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\to q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)=\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)$$
269	268	csbeq1d	$$⊢q=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
270	75	fconst	$$⊢\left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right):\left(m+1+1\dots N\right)⟶\left\{0\right\}$$
271	270	jctr	$$⊢q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\to \left(q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right):\left(m+1+1\dots N\right)⟶\left\{0\right\}\right)$$
272	266	nnred	$$⊢m\in {\mathbb{N}}_0\to m+1\in ℝ$$
273	272	ltp1d	$$⊢m\in {\mathbb{N}}_0\to m+1<m+1+1$$
274		fzdisj	$$⊢m+1<m+1+1\to \left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing$$
275	273 274	syl	$$⊢m\in {\mathbb{N}}_0\to \left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing$$
276		fun	$$⊢\left(\left(q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right):\left(m+1+1\dots N\right)⟶\left\{0\right\}\right)\wedge \left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing \right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)⟶\left(0\dots K\right)\cup \left\{0\right\}$$
277	271 275 276	syl2anr	$$⊢\left(m\in {\mathbb{N}}_0\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)⟶\left(0\dots K\right)\cup \left\{0\right\}$$
278	277	adantlr	$$⊢\left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)⟶\left(0\dots K\right)\cup \left\{0\right\}$$
279	278	adantl	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)⟶\left(0\dots K\right)\cup \left\{0\right\}$$
280	266	peano2nnd	$$⊢m\in {\mathbb{N}}_0\to m+1+1\in \mathbb{N}$$
281	280 192	eleqtrdi	$$⊢m\in {\mathbb{N}}_0\to m+1+1\in {\mathbb{Z}}_{\ge 1}$$
282	281	ad2antrl	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to m+1+1\in {\mathbb{Z}}_{\ge 1}$$
283		nn0z	$$⊢m\in {\mathbb{N}}_0\to m\in \mathbb{Z}$$
284	1	nnzd	$$⊢\phi \to N\in \mathbb{Z}$$
285		zltp1le	$$⊢\left(m\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(m<N\leftrightarrow m+1\le N\right)$$
286	283 284 285	syl2anr	$$⊢\left(\phi \wedge m\in {\mathbb{N}}_0\right)\to \left(m<N\leftrightarrow m+1\le N\right)$$
287	286	biimpa	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m<N\right)\to m+1\le N$$
288	287	anasss	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to m+1\le N$$
289	283	peano2zd	$$⊢m\in {\mathbb{N}}_0\to m+1\in \mathbb{Z}$$
290	289	adantr	$$⊢\left(m\in {\mathbb{N}}_0\wedge m<N\right)\to m+1\in \mathbb{Z}$$
291		eluz	$$⊢\left(m+1\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(N\in {\mathbb{Z}}_{\ge \left(m+1\right)}\leftrightarrow m+1\le N\right)$$
292	290 284 291	syl2anr	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(N\in {\mathbb{Z}}_{\ge \left(m+1\right)}\leftrightarrow m+1\le N\right)$$
293	288 292	mpbird	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to N\in {\mathbb{Z}}_{\ge \left(m+1\right)}$$
294		fzsplit2	$$⊢\left(m+1+1\in {\mathbb{Z}}_{\ge 1}\wedge N\in {\mathbb{Z}}_{\ge \left(m+1\right)}\right)\to \left(1\dots N\right)=\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)$$
295	282 293 294	syl2anc	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(1\dots N\right)=\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)$$
296	295	eqcomd	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)=\left(1\dots N\right)$$
297	196 197	syl	$$⊢\phi \to 0\in \left(0\dots K\right)$$
298	297	snssd	$$⊢\phi \to \left\{0\right\}\subseteq \left(0\dots K\right)$$
299		ssequn2	$$⊢\left\{0\right\}\subseteq \left(0\dots K\right)\leftrightarrow \left(0\dots K\right)\cup \left\{0\right\}=\left(0\dots K\right)$$
300	298 299	sylib	$$⊢\phi \to \left(0\dots K\right)\cup \left\{0\right\}=\left(0\dots K\right)$$
301	300	adantr	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(0\dots K\right)\cup \left\{0\right\}=\left(0\dots K\right)$$
302	296 301	feq23d	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)⟶\left(0\dots K\right)\cup \left\{0\right\}\leftrightarrow \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)$$
303	302	adantrr	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to \left(\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots m+1\right)\cup \left(m+1+1\dots N\right)⟶\left(0\dots K\right)\cup \left\{0\right\}\leftrightarrow \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)$$
304	279 303	mpbid	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
305		nfv	$$⊢Ⅎp\left(\phi \wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)$$
306		nfcsb1v	$$⊢\underset{\_}{Ⅎ}p⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
307	306	nfel1	$$⊢Ⅎp⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)$$
308	305 307	nfim	$$⊢Ⅎp\left(\left(\phi \wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)\right)$$
309		vex	$$⊢q\in \mathrm{V}$$
310		ovex	$$⊢\left(m+1+1\dots N\right)\in \mathrm{V}$$
311	310 208	xpex	$$⊢\left(m+1+1\dots N\right)\times \left\{0\right\}\in \mathrm{V}$$
312	309 311	unex	$$⊢q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\in \mathrm{V}$$
313		feq1	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(p:\left(1\dots N\right)⟶\left(0\dots K\right)\leftrightarrow \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)$$
314	313	anbi2d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\phi \wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\right)\leftrightarrow \left(\phi \wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)\right)$$
315		csbeq1a	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to B=⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B$$
316	315	eleq1d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(B\in \left(0\dots N\right)\leftrightarrow ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)\right)$$
317	314 316	imbi12d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\left(\phi \wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\right)\to B\in \left(0\dots N\right)\right)\leftrightarrow \left(\left(\phi \wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)\right)\right)$$
318	308 312 317 3	vtoclf	$$⊢\left(\phi \wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)$$
319	304 318	syldan	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)$$
320	319	anassrs	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)$$
321		elfznn0	$$⊢⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in {\mathbb{N}}_0$$
322	320 321	syl	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in {\mathbb{N}}_0$$
323	266	nnnn0d	$$⊢m\in {\mathbb{N}}_0\to m+1\in {\mathbb{N}}_0$$
324	323	adantr	$$⊢\left(m\in {\mathbb{N}}_0\wedge m<N\right)\to m+1\in {\mathbb{N}}_0$$
325	324	ad2antlr	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to m+1\in {\mathbb{N}}_0$$
326		leloe	$$⊢\left(m+1\in ℝ\wedge N\in ℝ\right)\to \left(m+1\le N\leftrightarrow \left(m+1<N\vee m+1=N\right)\right)$$
327	272 8 326	syl2anr	$$⊢\left(\phi \wedge m\in {\mathbb{N}}_0\right)\to \left(m+1\le N\leftrightarrow \left(m+1<N\vee m+1=N\right)\right)$$
328	286 327	bitrd	$$⊢\left(\phi \wedge m\in {\mathbb{N}}_0\right)\to \left(m<N\leftrightarrow \left(m+1<N\vee m+1=N\right)\right)$$
329	328	biimpd	$$⊢\left(\phi \wedge m\in {\mathbb{N}}_0\right)\to \left(m<N\to \left(m+1<N\vee m+1=N\right)\right)$$
330	329	imdistani	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m<N\right)\to \left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(m+1<N\vee m+1=N\right)\right)$$
331	330	anasss	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(m+1<N\vee m+1=N\right)\right)$$
332		simplll	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to \phi$$
333	280	nnge1d	$$⊢m\in {\mathbb{N}}_0\to 1\le m+1+1$$
334	333	ad2antlr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to 1\le m+1+1$$
335		zltp1le	$$⊢\left(m+1\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(m+1<N\leftrightarrow m+1+1\le N\right)$$
336	289 284 335	syl2anr	$$⊢\left(\phi \wedge m\in {\mathbb{N}}_0\right)\to \left(m+1<N\leftrightarrow m+1+1\le N\right)$$
337	336	biimpa	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m+1+1\le N$$
338	289	peano2zd	$$⊢m\in {\mathbb{N}}_0\to m+1+1\in \mathbb{Z}$$
339		1z	$$⊢1\in \mathbb{Z}$$
340		elfz	$$⊢\left(m+1+1\in \mathbb{Z}\wedge 1\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(m+1+1\in \left(1\dots N\right)\leftrightarrow \left(1\le m+1+1\wedge m+1+1\le N\right)\right)$$
341	339 340	mp3an2	$$⊢\left(m+1+1\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(m+1+1\in \left(1\dots N\right)\leftrightarrow \left(1\le m+1+1\wedge m+1+1\le N\right)\right)$$
342	338 284 341	syl2anr	$$⊢\left(\phi \wedge m\in {\mathbb{N}}_0\right)\to \left(m+1+1\in \left(1\dots N\right)\leftrightarrow \left(1\le m+1+1\wedge m+1+1\le N\right)\right)$$
343	342	adantr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to \left(m+1+1\in \left(1\dots N\right)\leftrightarrow \left(1\le m+1+1\wedge m+1+1\le N\right)\right)$$
344	334 337 343	mpbir2and	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m+1+1\in \left(1\dots N\right)$$
345	344	adantlr	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to m+1+1\in \left(1\dots N\right)$$
346		nn0re	$$⊢m\in {\mathbb{N}}_0\to m\in ℝ$$
347	346	ad2antlr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m\in ℝ$$
348	272	ad2antlr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m+1\in ℝ$$
349	8	ad2antrr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to N\in ℝ$$
350	346	ltp1d	$$⊢m\in {\mathbb{N}}_0\to m<m+1$$
351	350	ad2antlr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m<m+1$$
352		simpr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m+1<N$$
353	347 348 349 351 352	lttrd	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m<N$$
354	353	adantlr	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to m<N$$
355		anass	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m<N\right)\leftrightarrow \left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)$$
356	304	anassrs	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
357	355 356	sylanb	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m<N\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
358	357	an32s	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m<N\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
359	354 358	syldan	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
360		ffn	$$⊢q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\to qFn\left(1\dots m+1\right)$$
361	360	ad2antlr	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to qFn\left(1\dots m+1\right)$$
362	275	ad3antlr	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to \left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing$$
363		eluz	$$⊢\left(m+1+1\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(N\in {\mathbb{Z}}_{\ge \left(m+1+1\right)}\leftrightarrow m+1+1\le N\right)$$
364	338 284 363	syl2anr	$$⊢\left(\phi \wedge m\in {\mathbb{N}}_0\right)\to \left(N\in {\mathbb{Z}}_{\ge \left(m+1+1\right)}\leftrightarrow m+1+1\le N\right)$$
365	364	adantr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to \left(N\in {\mathbb{Z}}_{\ge \left(m+1+1\right)}\leftrightarrow m+1+1\le N\right)$$
366	337 365	mpbird	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to N\in {\mathbb{Z}}_{\ge \left(m+1+1\right)}$$
367		eluzfz1	$$⊢N\in {\mathbb{Z}}_{\ge \left(m+1+1\right)}\to m+1+1\in \left(m+1+1\dots N\right)$$
368	366 367	syl	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m+1<N\right)\to m+1+1\in \left(m+1+1\dots N\right)$$
369	368	adantlr	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to m+1+1\in \left(m+1+1\dots N\right)$$
370		fnconstg	$$⊢0\in \mathrm{V}\to \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)Fn\left(m+1+1\dots N\right)$$
371	75 370	ax-mp	$$⊢\left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)Fn\left(m+1+1\dots N\right)$$
372		fvun2	$$⊢\left(qFn\left(1\dots m+1\right)\wedge \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)Fn\left(m+1+1\dots N\right)\wedge \left(\left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing \wedge m+1+1\in \left(m+1+1\dots N\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=\left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\left(m+1+1\right)$$
373	371 372	mp3an2	$$⊢\left(qFn\left(1\dots m+1\right)\wedge \left(\left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing \wedge m+1+1\in \left(m+1+1\dots N\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=\left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\left(m+1+1\right)$$
374	361 362 369 373	syl12anc	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=\left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\left(m+1+1\right)$$
375	75	fvconst2	$$⊢m+1+1\in \left(m+1+1\dots N\right)\to \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\left(m+1+1\right)=0$$
376	369 375	syl	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\left(m+1+1\right)=0$$
377	374 376	eqtrd	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0$$
378		nfv	$$⊢Ⅎp\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0\right)\right)$$
379		nfcv	$$⊢\underset{\_}{Ⅎ}p<$$
380		nfcv	$$⊢\underset{\_}{Ⅎ}p\left(m+1+1\right)$$
381	306 379 380	nfbr	$$⊢Ⅎp⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1$$
382	378 381	nfim	$$⊢Ⅎp\left(\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1\right)$$
383		fveq1	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to p\left(m+1+1\right)=\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)$$
384	383	eqeq1d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(p\left(m+1+1\right)=0\leftrightarrow \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0\right)$$
385	313 384	3anbi23d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(m+1+1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(m+1+1\right)=0\right)\leftrightarrow \left(m+1+1\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0\right)\right)$$
386	385	anbi2d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(m+1+1\right)=0\right)\right)\leftrightarrow \left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0\right)\right)\right)$$
387	315	breq1d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(B<m+1+1\leftrightarrow ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1\right)$$
388	386 387	imbi12d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(m+1+1\right)=0\right)\right)\to B<m+1+1\right)\leftrightarrow \left(\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1\right)\right)$$
389		ovex	$$⊢m+1+1\in \mathrm{V}$$
390		eleq1	$$⊢n=m+1+1\to \left(n\in \left(1\dots N\right)\leftrightarrow m+1+1\in \left(1\dots N\right)\right)$$
391		fveqeq2	$$⊢n=m+1+1\to \left(p\left(n\right)=0\leftrightarrow p\left(m+1+1\right)=0\right)$$
392	390 391	3anbi13d	$$⊢n=m+1+1\to \left(\left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\leftrightarrow \left(m+1+1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(m+1+1\right)=0\right)\right)$$
393	392	anbi2d	$$⊢n=m+1+1\to \left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\leftrightarrow \left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(m+1+1\right)=0\right)\right)\right)$$
394		breq2	$$⊢n=m+1+1\to \left(B<n\leftrightarrow B<m+1+1\right)$$
395	393 394	imbi12d	$$⊢n=m+1+1\to \left(\left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\to B<n\right)\leftrightarrow \left(\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(m+1+1\right)=0\right)\right)\to B<m+1+1\right)\right)$$
396	389 395 4	vtocl	$$⊢\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(m+1+1\right)=0\right)\right)\to B<m+1+1$$
397	382 312 388 396	vtoclf	$$⊢\left(\phi \wedge \left(m+1+1\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(m+1+1\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1$$
398	332 345 359 377 397	syl13anc	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1$$
399	355 320	sylanb	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge m<N\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)$$
400	399	an32s	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m<N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)$$
401	400	elfzelzd	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m<N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \mathbb{Z}$$
402	354 401	syldan	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \mathbb{Z}$$
403	289	ad3antlr	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to m+1\in \mathbb{Z}$$
404		zleltp1	$$⊢\left(⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \mathbb{Z}\wedge m+1\in \mathbb{Z}\right)\to \left(⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1\leftrightarrow ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1\right)$$
405	402 403 404	syl2anc	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to \left(⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1\leftrightarrow ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<m+1+1\right)$$
406	398 405	mpbird	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1<N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1$$
407	350	ad2antlr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to m<m+1$$
408		breq2	$$⊢m+1=N\to \left(m<m+1\leftrightarrow m<N\right)$$
409	408	biimpac	$$⊢\left(m<m+1\wedge m+1=N\right)\to m<N$$
410	407 409	sylan	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1=N\right)\to m<N$$
411		elfzle2	$$⊢⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le N$$
412	400 411	syl	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m<N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le N$$
413	410 412	syldan	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1=N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le N$$
414		simpr	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1=N\right)\to m+1=N$$
415	413 414	breqtrrd	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge m+1=N\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1$$
416	406 415	jaodan	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\wedge \left(m+1<N\vee m+1=N\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1$$
417	416	an32s	$$⊢\left(\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(m+1<N\vee m+1=N\right)\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1$$
418	331 417	sylan	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1$$
419		elfz2nn0	$$⊢⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots m+1\right)\leftrightarrow \left(⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in {\mathbb{N}}_0\wedge m+1\in {\mathbb{N}}_0\wedge ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\le m+1\right)$$
420	322 325 418 419	syl3anbrc	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\in \left(0\dots m+1\right)$$
421		fzss2	$$⊢N\in {\mathbb{Z}}_{\ge \left(m+1\right)}\to \left(1\dots m+1\right)\subseteq \left(1\dots N\right)$$
422	293 421	syl	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(1\dots m+1\right)\subseteq \left(1\dots N\right)$$
423	422	sselda	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge n\in \left(1\dots m+1\right)\right)\to n\in \left(1\dots N\right)$$
424	423	3ad2antr1	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=0\right)\right)\to n\in \left(1\dots N\right)$$
425	356	3ad2antr2	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=0\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
426	360	ad2antll	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to qFn\left(1\dots m+1\right)$$
427	275	ad2antlr	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to \left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing$$
428		simprl	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to n\in \left(1\dots m+1\right)$$
429		fvun1	$$⊢\left(qFn\left(1\dots m+1\right)\wedge \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)Fn\left(m+1+1\dots N\right)\wedge \left(\left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing \wedge n\in \left(1\dots m+1\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=q\left(n\right)$$
430	371 429	mp3an2	$$⊢\left(qFn\left(1\dots m+1\right)\wedge \left(\left(1\dots m+1\right)\cap \left(m+1+1\dots N\right)=\varnothing \wedge n\in \left(1\dots m+1\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=q\left(n\right)$$
431	426 427 428 430	syl12anc	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=q\left(n\right)$$
432	431	adantlrr	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=q\left(n\right)$$
433	432	3adantr3	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=0\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=q\left(n\right)$$
434		simpr3	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=0\right)\right)\to q\left(n\right)=0$$
435	433 434	eqtrd	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=0\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0$$
436	424 425 435	3jca	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=0\right)\right)\to \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)$$
437		nfv	$$⊢Ⅎp\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)\right)$$
438		nfcv	$$⊢\underset{\_}{Ⅎ}pn$$
439	306 379 438	nfbr	$$⊢Ⅎp⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<n$$
440	437 439	nfim	$$⊢Ⅎp\left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<n\right)$$
441		fveq1	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to p\left(n\right)=\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)$$
442	441	eqeq1d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(p\left(n\right)=0\leftrightarrow \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)$$
443	313 442	3anbi23d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\leftrightarrow \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)\right)$$
444	443	anbi2d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\leftrightarrow \left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)\right)\right)$$
445	315	breq1d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(B<n\leftrightarrow ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<n\right)$$
446	444 445	imbi12d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=0\right)\right)\to B<n\right)\leftrightarrow \left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<n\right)\right)$$
447	440 312 446 4	vtoclf	$$⊢\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<n$$
448	447	adantlr	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<n$$
449	436 448	syldan	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=0\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B<n$$
450		simp1	$$⊢\left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\to n\in \left(1\dots m+1\right)$$
451	423	anasss	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge n\in \left(1\dots m+1\right)\right)\right)\to n\in \left(1\dots N\right)$$
452	450 451	sylanr2	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\right)\to n\in \left(1\dots N\right)$$
453		simp2	$$⊢\left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\to q:\left(1\dots m+1\right)⟶\left(0\dots K\right)$$
454	453 304	sylanr2	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)$$
455	431	3adantr3	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=q\left(n\right)$$
456		simpr3	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\to q\left(n\right)=K$$
457	455 456	eqtrd	$$⊢\left(\left(\phi \wedge m\in {\mathbb{N}}_0\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K$$
458	457	anasss	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K$$
459	458	adantrlr	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\right)\to \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K$$
460	452 454 459	3jca	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\right)\to \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)$$
461		nfv	$$⊢Ⅎp\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)\right)$$
462		nfcv	$$⊢\underset{\_}{Ⅎ}p\left(n-1\right)$$
463	306 462	nfne	$$⊢Ⅎp⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\ne n-1$$
464	461 463	nfim	$$⊢Ⅎp\left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\ne n-1\right)$$
465	441	eqeq1d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(p\left(n\right)=K\leftrightarrow \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)$$
466	313 465	3anbi23d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=K\right)\leftrightarrow \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)\right)$$
467	466	anbi2d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=K\right)\right)\leftrightarrow \left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)\right)\right)$$
468	315	neeq1d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(B\ne n-1\leftrightarrow ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\ne n-1\right)$$
469	467 468	imbi12d	$$⊢p=q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\to \left(\left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge p:\left(1\dots N\right)⟶\left(0\dots K\right)\wedge p\left(n\right)=K\right)\right)\to B\ne n-1\right)\leftrightarrow \left(\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\ne n-1\right)\right)$$
470	464 312 469 5	vtoclf	$$⊢\left(\phi \wedge \left(n\in \left(1\dots N\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right):\left(1\dots N\right)⟶\left(0\dots K\right)\wedge \left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)\left(n\right)=K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\ne n-1$$
471	460 470	syldan	$$⊢\left(\phi \wedge \left(\left(m\in {\mathbb{N}}_0\wedge m<N\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\ne n-1$$
472	471	anassrs	$$⊢\left(\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\wedge \left(n\in \left(1\dots m+1\right)\wedge q:\left(1\dots m+1\right)⟶\left(0\dots K\right)\wedge q\left(n\right)=K\right)\right)\to ⦋\left(q\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\ne n-1$$
473	267 269 420 449 472	poimirlem27	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to 2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\right)$$
474	267 269 420	poimirlem26	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to 2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
475		fzfi	$$⊢\left(0\dots m+1\right)\in \mathrm{Fin}$$
476		xpfi	$$⊢\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}\wedge \left(0\dots m+1\right)\in \mathrm{Fin}\right)\to \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\in \mathrm{Fin}$$
477	256 475 476	mp2an	$$⊢\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\in \mathrm{Fin}$$
478		rabfi	$$⊢\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\in \mathrm{Fin}\to \left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\in \mathrm{Fin}$$
479		hashcl	$$⊢\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\in \mathrm{Fin}\to \left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in {\mathbb{N}}_0$$
480	477 478 479	mp2b	$$⊢\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in {\mathbb{N}}_0$$
481	480	nn0zi	$$⊢\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}$$
482		zsubcl	$$⊢\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in \mathbb{Z}\right)\to \left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in \mathbb{Z}$$
483	481 264 482	mp2an	$$⊢\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in \mathbb{Z}$$
484		zsubcl	$$⊢\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}\right)\to \left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}$$
485	481 260 484	mp2an	$$⊢\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}$$
486		dvds2sub	$$⊢\left(2\in \mathbb{Z}\wedge \left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in \mathbb{Z}\wedge \left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}\right)\to \left(\left(2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\right)\wedge 2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)\to 2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|-\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)\right)$$
487	242 483 485 486	mp3an	$$⊢\left(2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\right)\wedge 2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)\to 2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|-\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
488	473 474 487	syl2anc	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to 2\parallel \left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|-\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
489	480	nn0cni	$$⊢\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in ℂ$$
490	263	nn0cni	$$⊢\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in ℂ$$
491	259	nn0cni	$$⊢\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in ℂ$$
492		nnncan1	$$⊢\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in ℂ\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in ℂ\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in ℂ\right)\to \left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|-\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|$$
493	489 490 491 492	mp3an	$$⊢\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|-\left(\left|\left\{t\in \left(\left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)\times \left(0\dots m+1\right)\right)|\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(\left(0\dots m+1\right)\setminus \left\{2^{nd}\left(t\right)\right\}\right)i=⦋1^{st}\left(t\right)/s⦌⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|$$
494	488 493	breqtrdi	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to 2\parallel \left(\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\right)$$
495		dvdssub2	$$⊢\left(\left(2\in \mathbb{Z}\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\in \mathbb{Z}\wedge \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\in \mathbb{Z}\right)\wedge 2\parallel \left(\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|-\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\right)\right)\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\right)$$
496	265 494 495	sylancr	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\right)$$
497		nn0cn	$$⊢m\in {\mathbb{N}}_0\to m\in ℂ$$
498		pncan1	$$⊢m\in ℂ\to m+1-1=m$$
499	497 498	syl	$$⊢m\in {\mathbb{N}}_0\to m+1-1=m$$
500	499	oveq2d	$$⊢m\in {\mathbb{N}}_0\to \left(0\dots m+1-1\right)=\left(0\dots m\right)$$
501	500	rexeqdv	$$⊢m\in {\mathbb{N}}_0\to \left(\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
502	500 501	raleqbidv	$$⊢m\in {\mathbb{N}}_0\to \left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right)$$
503	502	3anbi1d	$$⊢m\in {\mathbb{N}}_0\to \left(\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\leftrightarrow \left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right)$$
504	503	rabbidv	$$⊢m\in {\mathbb{N}}_0\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}=\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}$$
505	504	fveq2d	$$⊢m\in {\mathbb{N}}_0\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|$$
506	505	ad2antrl	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|$$
507	1	adantr	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to N\in \mathbb{N}$$
508	6	adantr	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to K\in \mathbb{N}$$
509		simprl	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to m\in {\mathbb{N}}_0$$
510		simprr	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to m<N$$
511	507 508 509 510	poimirlem4	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\approx \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}$$
512		fzfi	$$⊢\left(1\dots m\right)\in \mathrm{Fin}$$
513		mapfi	$$⊢\left(\left(0..^K\right)\in \mathrm{Fin}\wedge \left(1\dots m\right)\in \mathrm{Fin}\right)\to {\left(0..^K\right)}^{\left(1\dots m\right)}\in \mathrm{Fin}$$
514	15 512 513	mp2an	$$⊢{\left(0..^K\right)}^{\left(1\dots m\right)}\in \mathrm{Fin}$$
515		ovex	$$⊢\left(1\dots m\right)\in \mathrm{V}$$
516	515 515	mapval	$$⊢{\left(1\dots m\right)}^{\left(1\dots m\right)}=\left\{f|f:\left(1\dots m\right)⟶\left(1\dots m\right)\right\}$$
517		mapfi	$$⊢\left(\left(1\dots m\right)\in \mathrm{Fin}\wedge \left(1\dots m\right)\in \mathrm{Fin}\right)\to {\left(1\dots m\right)}^{\left(1\dots m\right)}\in \mathrm{Fin}$$
518	512 512 517	mp2an	$$⊢{\left(1\dots m\right)}^{\left(1\dots m\right)}\in \mathrm{Fin}$$
519	516 518	eqeltrri	$$⊢\left\{f|f:\left(1\dots m\right)⟶\left(1\dots m\right)\right\}\in \mathrm{Fin}$$
520		f1of	$$⊢f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\to f:\left(1\dots m\right)⟶\left(1\dots m\right)$$
521	520	ss2abi	$$⊢\left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\subseteq \left\{f|f:\left(1\dots m\right)⟶\left(1\dots m\right)\right\}$$
522		ssfi	$$⊢\left(\left\{f|f:\left(1\dots m\right)⟶\left(1\dots m\right)\right\}\in \mathrm{Fin}\wedge \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\subseteq \left\{f|f:\left(1\dots m\right)⟶\left(1\dots m\right)\right\}\right)\to \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\in \mathrm{Fin}$$
523	519 521 522	mp2an	$$⊢\left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\in \mathrm{Fin}$$
524		xpfi	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m\right)}\in \mathrm{Fin}\wedge \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\in \mathrm{Fin}\right)\to \left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\in \mathrm{Fin}$$
525	514 523 524	mp2an	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\in \mathrm{Fin}$$
526		rabfi	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\in \mathrm{Fin}\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\in \mathrm{Fin}$$
527	525 526	ax-mp	$$⊢\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\in \mathrm{Fin}$$
528		rabfi	$$⊢\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\in \mathrm{Fin}\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\in \mathrm{Fin}$$
529	256 528	ax-mp	$$⊢\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\in \mathrm{Fin}$$
530		hashen	$$⊢\left(\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\in \mathrm{Fin}\wedge \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\in \mathrm{Fin}\right)\to \left(\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\leftrightarrow \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\approx \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right)$$
531	527 529 530	mp2an	$$⊢\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\leftrightarrow \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\approx \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}$$
532	511 531	sylibr	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|$$
533	506 532	eqtr4d	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|$$
534	533	breq2d	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\left(\forall i\in \left(0\dots m+1-1\right)\exists j\in \left(0\dots m+1-1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\wedge 1^{st}\left(s\right)\left(m+1\right)=0\wedge 2^{nd}\left(s\right)\left(m+1\right)=m+1\right)\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
535	496 534	bitrd	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
536	535	biimpd	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\to 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
537	536	con3d	$$⊢\left(\phi \wedge \left(m\in {\mathbb{N}}_0\wedge m<N\right)\right)\to \left(\neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
538	537	expcom	$$⊢\left(m\in {\mathbb{N}}_0\wedge m<N\right)\to \left(\phi \to \left(\neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
539	538	a2d	$$⊢\left(m\in {\mathbb{N}}_0\wedge m<N\right)\to \left(\left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\to \left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
540	539	3adant1	$$⊢\left(N\in {\mathbb{N}}_0\wedge m\in {\mathbb{N}}_0\wedge m<N\right)\to \left(\left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m\right)}\right)\times \left\{f|f:\left(1\dots m\right)\underset{1-1\; onto}{⟶}\left(1\dots m\right)\right\}\right)|\forall i\in \left(0\dots m\right)\exists j\in \left(0\dots m\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\to \left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots m+1\right)}\right)\times \left\{f|f:\left(1\dots m+1\right)\underset{1-1\; onto}{⟶}\left(1\dots m+1\right)\right\}\right)|\forall i\in \left(0\dots m+1\right)\exists j\in \left(0\dots m+1\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots m+1\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(m+1+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)\right)$$
541	112 137 162 187 241 540	fnn0ind	$$⊢\left(N\in {\mathbb{N}}_0\wedge N\in {\mathbb{N}}_0\wedge N\le N\right)\to \left(\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
542	10 541	mpcom	$$⊢\phi \to \neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|$$
543		dvds0	$$⊢2\in \mathbb{Z}\to 2\parallel 0$$
544	242 543	ax-mp	$$⊢2\parallel 0$$
545		hash0	$$⊢\left|\varnothing \right|=0$$
546	544 545	breqtrri	$$⊢2\parallel \left|\varnothing \right|$$
547		fveq2	$$⊢\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}=\varnothing \to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}\right|=\left|\varnothing \right|$$
548	546 547	breqtrrid	$$⊢\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}=\varnothing \to 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}\right|$$
549	8	ltp1d	$$⊢\phi \to N<N+1$$
550	284	peano2zd	$$⊢\phi \to N+1\in \mathbb{Z}$$
551		fzn	$$⊢\left(N+1\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(N<N+1\leftrightarrow \left(N+1\dots N\right)=\varnothing \right)$$
552	550 284 551	syl2anc	$$⊢\phi \to \left(N<N+1\leftrightarrow \left(N+1\dots N\right)=\varnothing \right)$$
553	549 552	mpbid	$$⊢\phi \to \left(N+1\dots N\right)=\varnothing$$
554	553	xpeq1d	$$⊢\phi \to \left(N+1\dots N\right)\times \left\{0\right\}=\varnothing \times \left\{0\right\}$$
555	554 91	eqtrdi	$$⊢\phi \to \left(N+1\dots N\right)\times \left\{0\right\}=\varnothing$$
556	555	uneq2d	$$⊢\phi \to \left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)=\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \varnothing$$
557		un0	$$⊢\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \varnothing =1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)$$
558	556 557	eqtrdi	$$⊢\phi \to \left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)=1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)$$
559	558	csbeq1d	$$⊢\phi \to ⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=⦋\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)/p⦌B$$
560		ovex	$$⊢1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\in \mathrm{V}$$
561	560 2	csbie	$$⊢⦋\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)/p⦌B=C$$
562	559 561	eqtrdi	$$⊢\phi \to ⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B=C$$
563	562	eqeq2d	$$⊢\phi \to \left(i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow i=C\right)$$
564	563	rexbidv	$$⊢\phi \to \left(\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \exists j\in \left(0\dots N\right)i=C\right)$$
565	564	ralbidv	$$⊢\phi \to \left(\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\leftrightarrow \forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right)$$
566	565	rabbidv	$$⊢\phi \to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}=\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}$$
567	566	fveq2d	$$⊢\phi \to \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|=\left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}\right|$$
568	567	breq2d	$$⊢\phi \to \left(2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\leftrightarrow 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}\right|\right)$$
569	548 568	syl5ibr	$$⊢\phi \to \left(\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}=\varnothing \to 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\right)$$
570	569	necon3bd	$$⊢\phi \to \left(\neg 2\parallel \left|\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=⦋\left(\left(1^{st}\left(s\right){+}_f\left(\left(2^{nd}\left(s\right)\left[\left(1\dots j\right)\right]\times \left\{1\right\}\right)\cup \left(2^{nd}\left(s\right)\left[\left(j+1\dots N\right)\right]\times \left\{0\right\}\right)\right)\right)\cup \left(\left(N+1\dots N\right)\times \left\{0\right\}\right)\right)/p⦌B\right\}\right|\to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}\ne \varnothing \right)$$
571	542 570	mpd	$$⊢\phi \to \left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}\ne \varnothing$$
572		rabn0	$$⊢\left\{s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)|\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C\right\}\ne \varnothing \leftrightarrow \exists s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C$$
573	571 572	sylib	$$⊢\phi \to \exists s\in \left(\left({\left(0..^K\right)}^{\left(1\dots N\right)}\right)\times \left\{f|f:\left(1\dots N\right)\underset{1-1\; onto}{⟶}\left(1\dots N\right)\right\}\right)\forall i\in \left(0\dots N\right)\exists j\in \left(0\dots N\right)i=C$$