Theorem cdleme20l2

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, penultimate line. D , F , Y , G represent s_2, f(s), t_2, f(t) respectively. (Contributed by NM, 20-Nov-2012)

	Ref	Expression
Hypotheses	cdleme19.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme19.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme19.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme19.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme19.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme19.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme19.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme19.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme19.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme19.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
	cdleme20.v	$$⊢V=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme20l2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Y\right)\in A$$

Proof

Step	Hyp	Ref	Expression
1		cdleme19.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme19.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme19.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme19.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme19.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme19.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme19.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme19.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme19.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		cdleme19.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
11		cdleme20.v	$$⊢V=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
12		simp11l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to K\in \mathrm{HL}$$
13	12	hllatd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to K\in \mathrm{Lat}$$
14		simp11r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to W\in H$$
15		simp12l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to P\in A$$
16		simp13l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to Q\in A$$
17		simp22l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to S\in A$$
18		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
19	1 2 3 4 5 6 7 18	cdleme1b	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge Q\in A\wedge S\in A\right)\right)\to F\in {\mathrm{Base}}_K$$
20	12 14 15 16 17 19	syl23anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to F\in {\mathrm{Base}}_K$$
21		simp21l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to R\in A$$
22	1 2 3 4 5 9 18	cdlemedb	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\right)\right)\to D\in {\mathrm{Base}}_K$$
23	12 14 21 17 22	syl22anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to D\in {\mathrm{Base}}_K$$
24		simp23l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to T\in A$$
25	1 2 3 4 5 6 8 18	cdleme1b	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge Q\in A\wedge T\in A\right)\right)\to G\in {\mathrm{Base}}_K$$
26	12 14 15 16 24 25	syl23anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to G\in {\mathrm{Base}}_K$$
27	1 2 3 4 5 10 18	cdlemedb	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge T\in A\right)\right)\to Y\in {\mathrm{Base}}_K$$
28	12 14 21 24 27	syl22anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to Y\in {\mathrm{Base}}_K$$
29	18 2	latj4	$$⊢\left(K\in \mathrm{Lat}\wedge \left(F\in {\mathrm{Base}}_K\wedge D\in {\mathrm{Base}}_K\right)\wedge \left(G\in {\mathrm{Base}}_K\wedge Y\in {\mathrm{Base}}_K\right)\right)\to \left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }\left(G\underset{\dot{}}{\vee }Y\right)=\left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }\left(D\underset{\dot{}}{\vee }Y\right)$$
30	13 20 23 26 28 29	syl122anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }\left(G\underset{\dot{}}{\vee }Y\right)=\left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }\left(D\underset{\dot{}}{\vee }Y\right)$$
31		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
32		simp22	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)$$
33		simp23	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)$$
34		simp21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)$$
35		simp31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(P\ne Q\wedge S\ne T\right)$$
36		simp321	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
37		simp322	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
38	36 37	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
39		simp323	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
40	1 2 3 4 5 6 7 8 9 10 11	cdleme20d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }Y\right)=V$$
41	31 32 33 34 35 38 39 40	syl133anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }Y\right)=V$$
42		simp22r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
43		simp31r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to S\ne T$$
44	1 2 3 4 5 11	lhpat2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge S\ne T\right)\right)\to V\in A$$
45	12 14 17 42 24 43 44	syl222anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to V\in A$$
46	41 45	eqeltrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }Y\right)\in A$$
47		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
48		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
49		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
50		simp31l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to P\ne Q$$
51	1 2 3 4 5 6 7	cdleme3fa	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to F\in A$$
52	47 48 49 32 50 36 51	syl132anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to F\in A$$
53	1 2 3 4 5 6 8	cdleme3fa	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to G\in A$$
54	47 48 49 33 50 37 53	syl132anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to G\in A$$
55		simp33r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
56	1 2 3 4 5 6 7 8	cdleme16b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to F\ne G$$
57	31 32 33 35 36 37 55 56	syl133anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to F\ne G$$
58		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
59	2 4 58	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge F\in A\wedge G\in A\right)\wedge F\ne G\right)\to F\underset{\dot{}}{\vee }G\in \mathrm{LLines}\left(K\right)$$
60	12 52 54 57 59	syl31anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to F\underset{\dot{}}{\vee }G\in \mathrm{LLines}\left(K\right)$$
61	1 2 3 4 5 9	cdlemeda	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to D\in A$$
62	12 14 17 42 21 39 36 61	syl223anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to D\in A$$
63		simp23r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \neg T\underset{\dot{}}{\le }W$$
64	1 2 3 4 5 10	cdlemeda	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Y\in A$$
65	12 14 24 63 21 39 37 64	syl223anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to Y\in A$$
66		simp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
67		simp33l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
68	1 2 3 4 5 6 7 8 9 10 11	cdleme20j	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to D\ne Y$$
69	47 48 49 34 32 33 35 66 67 68	syl333anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to D\ne Y$$
70	2 4 58	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge D\in A\wedge Y\in A\right)\wedge D\ne Y\right)\to D\underset{\dot{}}{\vee }Y\in \mathrm{LLines}\left(K\right)$$
71	12 62 65 69 70	syl31anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to D\underset{\dot{}}{\vee }Y\in \mathrm{LLines}\left(K\right)$$
72		eqid	$$⊢\mathrm{LPlanes}\left(K\right)=\mathrm{LPlanes}\left(K\right)$$
73	2 3 4 58 72	2llnmj	$$⊢\left(K\in \mathrm{HL}\wedge F\underset{\dot{}}{\vee }G\in \mathrm{LLines}\left(K\right)\wedge D\underset{\dot{}}{\vee }Y\in \mathrm{LLines}\left(K\right)\right)\to \left(\left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }Y\right)\in A\leftrightarrow \left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }\left(D\underset{\dot{}}{\vee }Y\right)\in \mathrm{LPlanes}\left(K\right)\right)$$
74	12 60 71 73	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }Y\right)\in A\leftrightarrow \left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }\left(D\underset{\dot{}}{\vee }Y\right)\in \mathrm{LPlanes}\left(K\right)\right)$$
75	46 74	mpbid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }\left(D\underset{\dot{}}{\vee }Y\right)\in \mathrm{LPlanes}\left(K\right)$$
76	30 75	eqeltrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }\left(G\underset{\dot{}}{\vee }Y\right)\in \mathrm{LPlanes}\left(K\right)$$
77	1 2 3 4 5 6 7 8 9 10 11	cdleme20l1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to F\underset{\dot{}}{\vee }D\in \mathrm{LLines}\left(K\right)$$
78	47 48 49 21 17 42 50 36 39 77	syl333anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to F\underset{\dot{}}{\vee }D\in \mathrm{LLines}\left(K\right)$$
79		eqid	$$⊢\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W=\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
80	1 2 3 4 5 6 8 7 10 9 79	cdleme20l1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to G\underset{\dot{}}{\vee }Y\in \mathrm{LLines}\left(K\right)$$
81	47 48 49 21 24 63 50 37 39 80	syl333anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to G\underset{\dot{}}{\vee }Y\in \mathrm{LLines}\left(K\right)$$
82	2 3 4 58 72	2llnmj	$$⊢\left(K\in \mathrm{HL}\wedge F\underset{\dot{}}{\vee }D\in \mathrm{LLines}\left(K\right)\wedge G\underset{\dot{}}{\vee }Y\in \mathrm{LLines}\left(K\right)\right)\to \left(\left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Y\right)\in A\leftrightarrow \left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }\left(G\underset{\dot{}}{\vee }Y\right)\in \mathrm{LPlanes}\left(K\right)\right)$$
83	12 78 81 82	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Y\right)\in A\leftrightarrow \left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }\left(G\underset{\dot{}}{\vee }Y\right)\in \mathrm{LPlanes}\left(K\right)\right)$$
84	76 83	mpbird	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne T\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(F\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Y\right)\in A$$