Theorem rlocaddval

Description: Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025)

	Ref	Expression
Hypotheses	rlocaddval.1	$$⊢B={\mathrm{Base}}_R$$
	rlocaddval.2	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	rlocaddval.3	$$⊢\underset{\dot{}}{+}={+}_R$$
	rlocaddval.4	$$⊢L=R\mathrm{RLocal}S$$
	rlocaddval.5	$$⊢\underset{\dot{}}{\sim }=R{~}_{\mathrm{RL}}S$$
	rlocaddval.r	$$⊢\phi \to R\in \mathrm{CRing}$$
	rlocaddval.s	$$⊢\phi \to S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)$$
	rlocaddval.6	$$⊢\phi \to E\in B$$
	rlocaddval.7	$$⊢\phi \to F\in B$$
	rlocaddval.8	$$⊢\phi \to G\in S$$
	rlocaddval.9	$$⊢\phi \to H\in S$$
	rlocaddval.10	$$⊢\underset{\dot{}}{\oplus }={+}_L$$
Assertion	rlocaddval	$$⊢\phi \to \left[⟨E,G⟩\right]\underset{\dot{}}{\sim }\underset{\dot{}}{\oplus }\left[⟨F,H⟩\right]\underset{\dot{}}{\sim }=\left[⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩\right]\underset{\dot{}}{\sim }$$

Proof

Step	Hyp	Ref	Expression
1		rlocaddval.1	$$⊢B={\mathrm{Base}}_R$$
2		rlocaddval.2	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		rlocaddval.3	$$⊢\underset{\dot{}}{+}={+}_R$$
4		rlocaddval.4	$$⊢L=R\mathrm{RLocal}S$$
5		rlocaddval.5	$$⊢\underset{\dot{}}{\sim }=R{~}_{\mathrm{RL}}S$$
6		rlocaddval.r	$$⊢\phi \to R\in \mathrm{CRing}$$
7		rlocaddval.s	$$⊢\phi \to S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)$$
8		rlocaddval.6	$$⊢\phi \to E\in B$$
9		rlocaddval.7	$$⊢\phi \to F\in B$$
10		rlocaddval.8	$$⊢\phi \to G\in S$$
11		rlocaddval.9	$$⊢\phi \to H\in S$$
12		rlocaddval.10	$$⊢\underset{\dot{}}{\oplus }={+}_L$$
13	8 10	opelxpd	$$⊢\phi \to ⟨E,G⟩\in \left(B\times S\right)$$
14	9 11	opelxpd	$$⊢\phi \to ⟨F,H⟩\in \left(B\times S\right)$$
15		eqid	$$⊢0_R=0_R$$
16		eqid	$$⊢{-}_R={-}_R$$
17		eqid	$$⊢{\le }_R={\le }_R$$
18		eqid	$$⊢\mathrm{Scalar}\left(R\right)=\mathrm{Scalar}\left(R\right)$$
19		eqid	$$⊢{\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}={\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}$$
20		eqid	$$⊢{\cdot }_R={\cdot }_R$$
21		eqid	$$⊢B\times S=B\times S$$
22		eqid	$$⊢\mathrm{TopSet}\left(R\right)=\mathrm{TopSet}\left(R\right)$$
23		eqid	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(R\right)$$
24		eqid	$$⊢\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)=\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)$$
25		eqid	$$⊢\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)=\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)$$
26		eqid	$$⊢\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)=\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)$$
27		eqid	$$⊢\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}=\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}$$
28		eqid	$$⊢\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)=\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)$$
29		eqid	$$⊢{\mathrm{mulGrp}}_R={\mathrm{mulGrp}}_R$$
30	29 1	mgpbas	$$⊢B={\mathrm{Base}}_{{\mathrm{mulGrp}}_R}$$
31	30	submss	$$⊢S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)\to S\subseteq B$$
32	7 31	syl	$$⊢\phi \to S\subseteq B$$
33	1 15 2 16 3 17 18 19 20 21 5 22 23 24 25 26 27 28 6 32	rlocval	$$⊢\phi \to R\mathrm{RLocal}S=\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right){/}_{𝑠}\underset{\dot{}}{\sim }$$
34	4 33	eqtrid	$$⊢\phi \to L=\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right){/}_{𝑠}\underset{\dot{}}{\sim }$$
35		eqidd	$$⊢\phi \to \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}$$
36		eqid	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}$$
37	36	imasvalstr	$$⊢\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)\mathrm{Struct}⟨1,12⟩$$
38		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
39		snsstp1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}$$
40		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}$$
41		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}$$
42	40 41	sstri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}$$
43	39 42	sstri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}$$
44	1	fvexi	$$⊢B\in \mathrm{V}$$
45	44	a1i	$$⊢\phi \to B\in \mathrm{V}$$
46	45 7	xpexd	$$⊢\phi \to B\times S\in \mathrm{V}$$
47		eqid	$$⊢{\mathrm{Base}}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}={\mathrm{Base}}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}$$
48	35 37 38 43 46 47	strfv3	$$⊢\phi \to {\mathrm{Base}}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}=B\times S$$
49	48	eqcomd	$$⊢\phi \to B\times S={\mathrm{Base}}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}$$
50		eqid	$$⊢1_R=1_R$$
51	1 15 50 2 16 21 5 6 7	erler	$$⊢\phi \to \underset{\dot{}}{\sim }\mathrm{Er}\left(B\times S\right)$$
52		tpex	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\in \mathrm{V}$$
53		tpex	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\in \mathrm{V}$$
54	52 53	unex	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\in \mathrm{V}$$
55		tpex	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\in \mathrm{V}$$
56	54 55	unex	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\in \mathrm{V}$$
57	56	a1i	$$⊢\phi \to \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\in \mathrm{V}$$
58	32	ad2antrr	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to S\subseteq B$$
59	58	ad2antrr	$$⊢\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\to S\subseteq B$$
60	59	ad2antrr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to S\subseteq B$$
61		eqidd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to ⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩=⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩$$
62		eqidd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to ⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩=⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
63	6	crnggrpd	$$⊢\phi \to R\in \mathrm{Grp}$$
64	63	ad6antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to R\in \mathrm{Grp}$$
65	6	crngringd	$$⊢\phi \to R\in \mathrm{Ring}$$
66	65	ad6antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to R\in \mathrm{Ring}$$
67		simplr	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to u\underset{\dot{}}{\sim }p$$
68	1 5 58 67	erlcl1	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to u\in \left(B\times S\right)$$
69	68	ad4antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to u\in \left(B\times S\right)$$
70		xp1st	$$⊢u\in \left(B\times S\right)\to 1^{st}\left(u\right)\in B$$
71	69 70	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(u\right)\in B$$
72		simpr	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to v\underset{\dot{}}{\sim }q$$
73	1 5 58 72	erlcl1	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to v\in \left(B\times S\right)$$
74	73	ad4antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to v\in \left(B\times S\right)$$
75		xp2nd	$$⊢v\in \left(B\times S\right)\to 2^{nd}\left(v\right)\in S$$
76	74 75	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(v\right)\in S$$
77	60 76	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(v\right)\in B$$
78	1 2 66 71 77	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\in B$$
79		xp1st	$$⊢v\in \left(B\times S\right)\to 1^{st}\left(v\right)\in B$$
80	74 79	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(v\right)\in B$$
81		xp2nd	$$⊢u\in \left(B\times S\right)\to 2^{nd}\left(u\right)\in S$$
82	69 81	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(u\right)\in S$$
83	60 82	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(u\right)\in B$$
84	1 2 66 80 83	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\in B$$
85	1 3 64 78 84	grpcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\in B$$
86	1 5 58 67	erlcl2	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to p\in \left(B\times S\right)$$
87	86	ad4antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to p\in \left(B\times S\right)$$
88		xp1st	$$⊢p\in \left(B\times S\right)\to 1^{st}\left(p\right)\in B$$
89	87 88	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(p\right)\in B$$
90	1 5 58 72	erlcl2	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to q\in \left(B\times S\right)$$
91	90	ad4antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to q\in \left(B\times S\right)$$
92		xp2nd	$$⊢q\in \left(B\times S\right)\to 2^{nd}\left(q\right)\in S$$
93	91 92	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(q\right)\in S$$
94	60 93	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(q\right)\in B$$
95	1 2 66 89 94	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in B$$
96		xp1st	$$⊢q\in \left(B\times S\right)\to 1^{st}\left(q\right)\in B$$
97	91 96	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(q\right)\in B$$
98		xp2nd	$$⊢p\in \left(B\times S\right)\to 2^{nd}\left(p\right)\in S$$
99	87 98	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(p\right)\in S$$
100	60 99	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(p\right)\in B$$
101	1 2 66 97 100	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\in B$$
102	1 3 64 95 101	grpcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\in B$$
103	7	ad6antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)$$
104	29 2	mgpplusg	$$⊢\underset{\dot{}}{·}={+}_{{\mathrm{mulGrp}}_R}$$
105	104	submcl	$$⊢\left(S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)\wedge 2^{nd}\left(u\right)\in S\wedge 2^{nd}\left(v\right)\in S\right)\to 2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\in S$$
106	103 82 76 105	syl3anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\in S$$
107	104	submcl	$$⊢\left(S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)\wedge 2^{nd}\left(p\right)\in S\wedge 2^{nd}\left(q\right)\in S\right)\to 2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in S$$
108	103 99 93 107	syl3anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in S$$
109		simp-4r	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\in S$$
110		simplr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\in S$$
111	104	submcl	$$⊢\left(S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)\wedge f\in S\wedge g\in S\right)\to f\underset{\dot{}}{·}g\in S$$
112	103 109 110 111	syl3anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}g\in S$$
113	60 108	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in B$$
114	1 3 2	ringdir	$$⊢\left(R\in \mathrm{Ring}\wedge \left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\in B\wedge 1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\in B\wedge 2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in B\right)\right)\to \left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)=\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)$$
115	66 78 84 113 114	syl13anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)=\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)$$
116	60 106	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\in B$$
117	1 3 2	ringdir	$$⊢\left(R\in \mathrm{Ring}\wedge \left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in B\wedge 1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\in B\wedge 2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\in B\right)\right)\to \left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)=\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)$$
118	66 95 101 116 117	syl13anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)=\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)$$
119	115 118	oveq12d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right){-}_R\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)$$
120	119	oveq2d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right){-}_R\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)$$
121	60 109	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\in B$$
122	60 110	sseldd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\in B$$
123	1 2 66 121 122	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}g\in B$$
124	1 2 66 78 113	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\in B$$
125	1 2 66 84 113	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\in B$$
126	1 3 64 124 125	grpcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\in B$$
127	1 2 66 95 116	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\in B$$
128	1 2 66 101 116	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\in B$$
129	1 3 64 127 128	grpcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\in B$$
130	1 2 16 66 123 126 129	ringsubdi	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)=\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)$$
131	1 3 2	ringdi	$$⊢\left(R\in \mathrm{Ring}\wedge \left(f\underset{\dot{}}{·}g\in B\wedge \left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\in B\wedge \left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\in B\right)\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)=\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)$$
132	66 123 124 125 131	syl13anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)=\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)$$
133	1 3 2	ringdi	$$⊢\left(R\in \mathrm{Ring}\wedge \left(f\underset{\dot{}}{·}g\in B\wedge \left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\in B\wedge \left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\in B\right)\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)$$
134	66 123 127 128 133	syl13anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)$$
135	132 134	oveq12d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)=\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\right){-}_R\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)$$
136	66	ringabld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to R\in \mathrm{Abel}$$
137	1 2 66 123 124	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\in B$$
138	1 2 66 123 125	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\in B$$
139	1 2 66 123 127	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\in B$$
140	1 2 66 123 128	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\in B$$
141	1 3 16	ablsub4	$$⊢\left(R\in \mathrm{Abel}\wedge \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\in B\wedge \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\in B\right)\wedge \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\in B\wedge \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\in B\right)\right)\to \left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\right){-}_R\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)=\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)$$
142	136 137 138 139 140 141	syl122anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\right){-}_R\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)=\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)$$
143	29	crngmgp	$$⊢R\in \mathrm{CRing}\to {\mathrm{mulGrp}}_R\in \mathrm{CMnd}$$
144	6 143	syl	$$⊢\phi \to {\mathrm{mulGrp}}_R\in \mathrm{CMnd}$$
145	144	ad6antr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to {\mathrm{mulGrp}}_R\in \mathrm{CMnd}$$
146	30 104 145 121 122 71 77 100 94	cmn246135	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)=\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)$$
147	30 104 145 121 122 89 94 83 77	cmn246135	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=\left(g\underset{\dot{}}{·}\left(2^{nd}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)$$
148	30 104	cmncom	$$⊢\left({\mathrm{mulGrp}}_R\in \mathrm{CMnd}\wedge 2^{nd}\left(v\right)\in B\wedge 2^{nd}\left(q\right)\in B\right)\to 2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)=2^{nd}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)$$
149	145 77 94 148	syl3anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)=2^{nd}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)$$
150	149	oveq2d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)=g\underset{\dot{}}{·}\left(2^{nd}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)$$
151	150	oveq1d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=\left(g\underset{\dot{}}{·}\left(2^{nd}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)$$
152	147 151	eqtr4d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)$$
153	146 152	oveq12d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=\left(\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\right){-}_R\left(\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\right)$$
154	1 2 66 71 100	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\in B$$
155	1 2 66 89 83	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\in B$$
156	1 2 16 66 121 154 155	ringsubdi	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=\left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right){-}_R\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)$$
157		simpllr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R$$
158	156 157	eqtr3d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right){-}_R\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R$$
159	158	oveq2d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(\left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right){-}_R\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\right)=\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}{0}_R$$
160	1 2 66 77 94	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in B$$
161	1 2 66 122 160	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\in B$$
162	1 2 66 121 154	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\in B$$
163	1 2 66 121 155	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\in B$$
164	1 2 16 66 161 162 163	ringsubdi	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(\left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right){-}_R\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\right)=\left(\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\right){-}_R\left(\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\right)$$
165	1 2 15 66 161	ringrzd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}{0}_R=0_R$$
166	159 164 165	3eqtr3d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\right){-}_R\left(\left(g\underset{\dot{}}{·}\left(2^{nd}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{·}\left(f\underset{\dot{}}{·}\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\right)=0_R$$
167	153 166	eqtrd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=0_R$$
168	30 104 145 121 122 80 83 100 94	cmn145236	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)=\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)$$
169	30 104 145 121 122 97 100 83 77	cmn145236	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=\left(f\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)$$
170	30 104	cmncom	$$⊢\left({\mathrm{mulGrp}}_R\in \mathrm{CMnd}\wedge 2^{nd}\left(p\right)\in B\wedge 2^{nd}\left(u\right)\in B\right)\to 2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)=2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)$$
171	145 100 83 170	syl3anc	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)=2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)$$
172	171	oveq2d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)=f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)$$
173	172	oveq1d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)$$
174	169 173	eqtrd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)$$
175	168 174	oveq12d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=\left(\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)$$
176	1 2 66 80 94	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in B$$
177	1 2 66 97 77	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\in B$$
178	1 2 16 66 122 176 177	ringsubdi	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=\left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right){-}_R\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)$$
179		simpr	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R$$
180	178 179	eqtr3d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right){-}_R\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R$$
181	180	oveq2d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(\left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right){-}_R\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}{0}_R$$
182	1 2 66 83 100	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\in B$$
183	1 2 66 121 182	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\in B$$
184	1 2 66 122 176	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\in B$$
185	1 2 66 122 177	ringcld	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\in B$$
186	1 2 16 66 183 184 185	ringsubdi	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(\left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right){-}_R\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=\left(\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)$$
187	1 2 15 66 183	ringrzd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}{0}_R=0_R$$
188	181 186 187	3eqtr3d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(g\underset{\dot{}}{·}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=0_R$$
189	175 188	eqtrd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=0_R$$
190	167 189	oveq12d	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)=0_R\underset{\dot{}}{+}{0}_R$$
191	1 15	grpidcl	$$⊢R\in \mathrm{Grp}\to 0_R\in B$$
192	64 191	syl	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 0_R\in B$$
193	1 3 15 64 192	grplidd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to 0_R\underset{\dot{}}{+}{0}_R=0_R$$
194	190 193	eqtrd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)\underset{\dot{}}{+}\left(\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)=0_R$$
195	135 142 194	3eqtrd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right)\right)\right){-}_R\left(\left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\underset{\dot{}}{+}\left(\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)\right)=0_R$$
196	120 130 195	3eqtrd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to \left(f\underset{\dot{}}{·}g\right)\underset{\dot{}}{·}\left(\left(\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\right){-}_R\left(\left(\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\right)\underset{\dot{}}{·}\left(2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)\right)=0_R$$
197	1 5 60 15 2 16 61 62 85 102 106 108 112 196	erlbrd	$$⊢\left(\left(\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\wedge g\in S\right)\wedge g\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R\right)\to ⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩\underset{\dot{}}{\sim }⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
198	72	ad2antrr	$$⊢\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\to v\underset{\dot{}}{\sim }q$$
199	1 5 59 15 2 16 198	erldi	$$⊢\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\to \exists g\in Sg\underset{\dot{}}{·}\left(\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right){-}_R\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\right)=0_R$$
200	197 199	r19.29a	$$⊢\left(\left(\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\wedge f\in S\right)\wedge f\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R\right)\to ⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩\underset{\dot{}}{\sim }⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
201	1 5 58 15 2 16 67	erldi	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to \exists f\in Sf\underset{\dot{}}{·}\left(\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right){-}_R\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)\right)=0_R$$
202	200 201	r19.29a	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to ⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩\underset{\dot{}}{\sim }⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
203		plusgid	$$⊢{+}_{𝑔}=\mathrm{Slot}{+}_{\mathrm{ndx}}$$
204		snsstp2	$$⊢\left\{⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}$$
205	204 42	sstri	$$⊢\left\{⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}$$
206	24	mpoexg	$$⊢\left(B\times S\in \mathrm{V}\wedge B\times S\in \mathrm{V}\right)\to \left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)\in \mathrm{V}$$
207	46 46 206	syl2anc	$$⊢\phi \to \left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)\in \mathrm{V}$$
208		eqid	$$⊢{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}={+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}$$
209	35 37 203 205 207 208	strfv3	$$⊢\phi \to {+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}=\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)$$
210	209	ad2antrr	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to {+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}=\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)$$
211	210	oveqd	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to u{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}v=u\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)v$$
212		opex	$$⊢⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩\in \mathrm{V}$$
213	212	a1i	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to ⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩\in \mathrm{V}$$
214		simpl	$$⊢\left(a=u\wedge b=v\right)\to a=u$$
215	214	fveq2d	$$⊢\left(a=u\wedge b=v\right)\to 1^{st}\left(a\right)=1^{st}\left(u\right)$$
216		simpr	$$⊢\left(a=u\wedge b=v\right)\to b=v$$
217	216	fveq2d	$$⊢\left(a=u\wedge b=v\right)\to 2^{nd}\left(b\right)=2^{nd}\left(v\right)$$
218	215 217	oveq12d	$$⊢\left(a=u\wedge b=v\right)\to 1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)=1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)$$
219	216	fveq2d	$$⊢\left(a=u\wedge b=v\right)\to 1^{st}\left(b\right)=1^{st}\left(v\right)$$
220	214	fveq2d	$$⊢\left(a=u\wedge b=v\right)\to 2^{nd}\left(a\right)=2^{nd}\left(u\right)$$
221	219 220	oveq12d	$$⊢\left(a=u\wedge b=v\right)\to 1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)=1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)$$
222	218 221	oveq12d	$$⊢\left(a=u\wedge b=v\right)\to \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)=\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right)$$
223	220 217	oveq12d	$$⊢\left(a=u\wedge b=v\right)\to 2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)=2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)$$
224	222 223	opeq12d	$$⊢\left(a=u\wedge b=v\right)\to ⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩=⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩$$
225	224 24	ovmpoga	$$⊢\left(u\in \left(B\times S\right)\wedge v\in \left(B\times S\right)\wedge ⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩\in \mathrm{V}\right)\to u\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)v=⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩$$
226	68 73 213 225	syl3anc	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to u\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)v=⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩$$
227	211 226	eqtrd	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to u{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}v=⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩$$
228	210	oveqd	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q=p\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)q$$
229		opex	$$⊢⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩\in \mathrm{V}$$
230	229	a1i	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to ⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩\in \mathrm{V}$$
231		simpl	$$⊢\left(a=p\wedge b=q\right)\to a=p$$
232	231	fveq2d	$$⊢\left(a=p\wedge b=q\right)\to 1^{st}\left(a\right)=1^{st}\left(p\right)$$
233		simpr	$$⊢\left(a=p\wedge b=q\right)\to b=q$$
234	233	fveq2d	$$⊢\left(a=p\wedge b=q\right)\to 2^{nd}\left(b\right)=2^{nd}\left(q\right)$$
235	232 234	oveq12d	$$⊢\left(a=p\wedge b=q\right)\to 1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)=1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)$$
236	233	fveq2d	$$⊢\left(a=p\wedge b=q\right)\to 1^{st}\left(b\right)=1^{st}\left(q\right)$$
237	231	fveq2d	$$⊢\left(a=p\wedge b=q\right)\to 2^{nd}\left(a\right)=2^{nd}\left(p\right)$$
238	236 237	oveq12d	$$⊢\left(a=p\wedge b=q\right)\to 1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)=1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)$$
239	235 238	oveq12d	$$⊢\left(a=p\wedge b=q\right)\to \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)=\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)$$
240	237 234	oveq12d	$$⊢\left(a=p\wedge b=q\right)\to 2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)=2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)$$
241	239 240	opeq12d	$$⊢\left(a=p\wedge b=q\right)\to ⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩=⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
242	241 24	ovmpoga	$$⊢\left(p\in \left(B\times S\right)\wedge q\in \left(B\times S\right)\wedge ⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩\in \mathrm{V}\right)\to p\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)q=⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
243	86 90 230 242	syl3anc	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to p\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)q=⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
244	228 243	eqtrd	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q=⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
245	227 244	breq12d	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to \left(\left(u{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}v\right)\underset{\dot{}}{\sim }\left(p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q\right)\leftrightarrow ⟨\left(1^{st}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(v\right)\underset{\dot{}}{·}{2}^{nd}\left(u\right)\right),2^{nd}\left(u\right)\underset{\dot{}}{·}{2}^{nd}\left(v\right)⟩\underset{\dot{}}{\sim }⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩\right)$$
246	202 245	mpbird	$$⊢\left(\left(\phi \wedge u\underset{\dot{}}{\sim }p\right)\wedge v\underset{\dot{}}{\sim }q\right)\to \left(u{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}v\right)\underset{\dot{}}{\sim }\left(p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q\right)$$
247	246	anasss	$$⊢\left(\phi \wedge \left(u\underset{\dot{}}{\sim }p\wedge v\underset{\dot{}}{\sim }q\right)\right)\to \left(u{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}v\right)\underset{\dot{}}{\sim }\left(p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q\right)$$
248	247	ex	$$⊢\phi \to \left(\left(u\underset{\dot{}}{\sim }p\wedge v\underset{\dot{}}{\sim }q\right)\to \left(u{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}v\right)\underset{\dot{}}{\sim }\left(p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q\right)\right)$$
249	209	oveqd	$$⊢\phi \to p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q=p\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)q$$
250	249	ad2antrr	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q=p\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)q$$
251		simplr	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to p\in \left(B\times S\right)$$
252		simpr	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to q\in \left(B\times S\right)$$
253	229	a1i	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to ⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩\in \mathrm{V}$$
254	251 252 253 242	syl3anc	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to p\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)q=⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩$$
255	63	ad2antrr	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to R\in \mathrm{Grp}$$
256	65	ad2antrr	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to R\in \mathrm{Ring}$$
257	251 88	syl	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 1^{st}\left(p\right)\in B$$
258	32	ad2antrr	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to S\subseteq B$$
259	252 92	syl	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 2^{nd}\left(q\right)\in S$$
260	258 259	sseldd	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 2^{nd}\left(q\right)\in B$$
261	1 2 256 257 260	ringcld	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in B$$
262	252 96	syl	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 1^{st}\left(q\right)\in B$$
263	251 98	syl	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 2^{nd}\left(p\right)\in S$$
264	258 263	sseldd	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 2^{nd}\left(p\right)\in B$$
265	1 2 256 262 264	ringcld	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\in B$$
266	1 3 255 261 265	grpcld	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to \left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right)\in B$$
267	7	ad2antrr	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to S\in \mathrm{SubMnd}\left({\mathrm{mulGrp}}_R\right)$$
268	267 263 259 107	syl3anc	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to 2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\in S$$
269	266 268	opelxpd	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to ⟨\left(1^{st}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(q\right)\underset{\dot{}}{·}{2}^{nd}\left(p\right)\right),2^{nd}\left(p\right)\underset{\dot{}}{·}{2}^{nd}\left(q\right)⟩\in \left(B\times S\right)$$
270	254 269	eqeltrd	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to p\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)q\in \left(B\times S\right)$$
271	250 270	eqeltrd	$$⊢\left(\left(\phi \wedge p\in \left(B\times S\right)\right)\wedge q\in \left(B\times S\right)\right)\to p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q\in \left(B\times S\right)$$
272	271	anasss	$$⊢\left(\phi \wedge \left(p\in \left(B\times S\right)\wedge q\in \left(B\times S\right)\right)\right)\to p{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}q\in \left(B\times S\right)$$
273	34 49 51 57 248 272 208 12	qusaddval	$$⊢\left(\phi \wedge ⟨E,G⟩\in \left(B\times S\right)\wedge ⟨F,H⟩\in \left(B\times S\right)\right)\to \left[⟨E,G⟩\right]\underset{\dot{}}{\sim }\underset{\dot{}}{\oplus }\left[⟨F,H⟩\right]\underset{\dot{}}{\sim }=\left[\left(⟨E,G⟩{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}⟨F,H⟩\right)\right]\underset{\dot{}}{\sim }$$
274	13 14 273	mpd3an23	$$⊢\phi \to \left[⟨E,G⟩\right]\underset{\dot{}}{\sim }\underset{\dot{}}{\oplus }\left[⟨F,H⟩\right]\underset{\dot{}}{\sim }=\left[\left(⟨E,G⟩{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}⟨F,H⟩\right)\right]\underset{\dot{}}{\sim }$$
275	209	oveqd	$$⊢\phi \to ⟨E,G⟩{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}⟨F,H⟩=⟨E,G⟩\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟨F,H⟩$$
276	24	a1i	$$⊢\phi \to \left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)=\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)$$
277		simprl	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to a=⟨E,G⟩$$
278	277	fveq2d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(a\right)=1^{st}\left(⟨E,G⟩\right)$$
279	8	adantr	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to E\in B$$
280	10	adantr	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to G\in S$$
281		op1stg	$$⊢\left(E\in B\wedge G\in S\right)\to 1^{st}\left(⟨E,G⟩\right)=E$$
282	279 280 281	syl2anc	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(⟨E,G⟩\right)=E$$
283	278 282	eqtrd	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(a\right)=E$$
284		simprr	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to b=⟨F,H⟩$$
285	284	fveq2d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 2^{nd}\left(b\right)=2^{nd}\left(⟨F,H⟩\right)$$
286	9	adantr	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to F\in B$$
287	11	adantr	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to H\in S$$
288		op2ndg	$$⊢\left(F\in B\wedge H\in S\right)\to 2^{nd}\left(⟨F,H⟩\right)=H$$
289	286 287 288	syl2anc	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 2^{nd}\left(⟨F,H⟩\right)=H$$
290	285 289	eqtrd	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 2^{nd}\left(b\right)=H$$
291	283 290	oveq12d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)=E\underset{\dot{}}{·}H$$
292	284	fveq2d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(b\right)=1^{st}\left(⟨F,H⟩\right)$$
293		op1stg	$$⊢\left(F\in B\wedge H\in S\right)\to 1^{st}\left(⟨F,H⟩\right)=F$$
294	286 287 293	syl2anc	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(⟨F,H⟩\right)=F$$
295	292 294	eqtrd	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(b\right)=F$$
296	277	fveq2d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 2^{nd}\left(a\right)=2^{nd}\left(⟨E,G⟩\right)$$
297		op2ndg	$$⊢\left(E\in B\wedge G\in S\right)\to 2^{nd}\left(⟨E,G⟩\right)=G$$
298	279 280 297	syl2anc	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 2^{nd}\left(⟨E,G⟩\right)=G$$
299	296 298	eqtrd	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 2^{nd}\left(a\right)=G$$
300	295 299	oveq12d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)=F\underset{\dot{}}{·}G$$
301	291 300	oveq12d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)=\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right)$$
302	299 290	oveq12d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to 2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)=G\underset{\dot{}}{·}H$$
303	301 302	opeq12d	$$⊢\left(\phi \wedge \left(a=⟨E,G⟩\wedge b=⟨F,H⟩\right)\right)\to ⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩=⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩$$
304		opex	$$⊢⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩\in \mathrm{V}$$
305	304	a1i	$$⊢\phi \to ⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩\in \mathrm{V}$$
306	276 303 13 14 305	ovmpod	$$⊢\phi \to ⟨E,G⟩\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟨F,H⟩=⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩$$
307	275 306	eqtrd	$$⊢\phi \to ⟨E,G⟩{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}⟨F,H⟩=⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩$$
308	307	eceq1d	$$⊢\phi \to \left[\left(⟨E,G⟩{+}_{\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B\times S⟩,⟨{+}_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\underset{\dot{}}{+}\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼⟨1^{st}\left(a\right)\underset{\dot{}}{·}{1}^{st}\left(b\right),2^{nd}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)⟩\right)⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(k\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},a\in \left(B\times S\right)⟼⟨k{\cdot }_R{1}^{st}\left(a\right),2^{nd}\left(a\right)⟩\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\varnothing ⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopSet}\left(R\right){\times }_t\left(\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}S\right)⟩,⟨{\le }_{\mathrm{ndx}},\left\{⟨a,b⟩|\left(\left(a\in \left(B\times S\right)\wedge b\in \left(B\times S\right)\right)\wedge \left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right){\le }_R\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)\right\}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(a\in \left(B\times S\right),b\in \left(B\times S\right)⟼\left(1^{st}\left(a\right)\underset{\dot{}}{·}{2}^{nd}\left(b\right)\right)\mathrm{dist}\left(R\right)\left(1^{st}\left(b\right)\underset{\dot{}}{·}{2}^{nd}\left(a\right)\right)\right)⟩\right\}\right)}⟨F,H⟩\right)\right]\underset{\dot{}}{\sim }=\left[⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩\right]\underset{\dot{}}{\sim }$$
309	274 308	eqtrd	$$⊢\phi \to \left[⟨E,G⟩\right]\underset{\dot{}}{\sim }\underset{\dot{}}{\oplus }\left[⟨F,H⟩\right]\underset{\dot{}}{\sim }=\left[⟨\left(E\underset{\dot{}}{·}H\right)\underset{\dot{}}{+}\left(F\underset{\dot{}}{·}G\right),G\underset{\dot{}}{·}H⟩\right]\underset{\dot{}}{\sim }$$