Theorem cdleme39n

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT. E , Y , G , Z serve as f(t), f(u), f_t( R ), f_t( S ). Put hypotheses of cdleme38n in convention of cdleme32sn1awN . TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013)

	Ref	Expression
Hypotheses	cdleme39.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme39.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme39.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme39.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme39.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme39.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme39.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme39.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme39.y	$$⊢Y=\left(u\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme39.z	$$⊢Z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme39n	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to G\ne Z$$

Proof

Step	Hyp	Ref	Expression
1		cdleme39.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme39.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme39.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme39.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme39.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme39.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme39.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme39.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme39.y	$$⊢Y=\left(u\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme39.z	$$⊢Z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		eqid	$$⊢\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W=\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W$$
12		eqid	$$⊢\left(u\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\wedge }W=\left(u\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\wedge }W$$
13		eqid	$$⊢\left(R\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(R\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14		eqid	$$⊢\left(S\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(S\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15	1 2 3 4 5 6 7 9 11 12 13 14	cdleme38n	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)\ne \left(S\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
16		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
17		simp12l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to P\in A$$
18		simp13l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to Q\in A$$
19		simp22l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R\in A$$
20		simp22r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg R\underset{\dot{}}{\le }W$$
21		simp311	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
22		simp32l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)$$
23	1 2 3 4 5 6 7 8 11	cdleme39a	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\right)\right)\to G=\left(R\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
24	16 17 18 19 20 21 22 23	syl322anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to G=\left(R\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
25		simp23l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to S\in A$$
26		simp23r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
27		simp312	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
28		simp33l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)$$
29	1 2 3 4 5 6 9 10 12	cdleme39a	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge P\in A\wedge Q\in A\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\right)\right)\to Z=\left(S\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
30	16 17 18 25 26 27 28 29	syl322anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to Z=\left(S\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
31	15 24 30	3netr4d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to G\ne Z$$