Theorem cdleme26eALTN

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 4th line on p. 115. F , N , O represent f(z), f_z(s), f_z(t) respectively. When t \/ v = p \/ q, f_z(s) <_ f_z(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme26eALT.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme26eALT.f	$$⊢F=\left(y\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }y\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme26eALT.g	$$⊢G=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme26eALT.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }y\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme26eALT.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }\left(\left(T\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme26eALT.i	$$⊢I=\left(\iota u\in B|\forall y\in A\left(\left(\neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
	cdleme26eALT.e	$$⊢E=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
Assertion	cdleme26eALTN	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to I\underset{\dot{}}{\le }\left(E\underset{\dot{}}{\vee }V\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme26eALT.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme26eALT.f	$$⊢F=\left(y\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }y\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme26eALT.g	$$⊢G=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme26eALT.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }y\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme26eALT.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }\left(\left(T\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
12		cdleme26eALT.i	$$⊢I=\left(\iota u\in B|\forall y\in A\left(\left(\neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
13		cdleme26eALT.e	$$⊢E=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
14		simp11l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to K\in \mathrm{HL}$$
15		simp11r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to W\in H$$
16		simp231	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to T\in A$$
17		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
18		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
19		simp21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to P\ne Q$$
20		simp221	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to S\in A$$
21		simp31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)$$
22		simp21	$$⊢\left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to y\in A$$
23	22	3ad2ant3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to y\in A$$
24		simp322	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg y\underset{\dot{}}{\le }W$$
25		simp31	$$⊢\left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to z\in A$$
26	25	3ad2ant3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to z\in A$$
27		simp332	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg z\underset{\dot{}}{\le }W$$
28	26 27	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)$$
29	23 24 28	jca31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)$$
30	2 3 4 5 6 7 8 9 10 11	cdleme22eALTN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\wedge T\in A\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(S\in A\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(\left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\right)\to N\underset{\dot{}}{\le }\left(O\underset{\dot{}}{\vee }V\right)$$
31	14 15 16 17 18 19 20 21 29 30	syl333anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to N\underset{\dot{}}{\le }\left(O\underset{\dot{}}{\vee }V\right)$$
32		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
33		simp222	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
34		simp223	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
35	1 2 3 4 5 6 7 8 10 12	cdleme25cl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to I\in B$$
36	32 17 18 20 33 19 34 35	syl322anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to I\in B$$
37		simp323	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
38	1	fvexi	$$⊢B\in \mathrm{V}$$
39	38 12	riotasv	$$⊢\left(I\in B\wedge y\in A\wedge \left(\neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to I=N$$
40	36 23 24 37 39	syl112anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to I=N$$
41		simp232	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg T\underset{\dot{}}{\le }W$$
42		simp233	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
43	1 2 3 4 5 6 7 9 11 13	cdleme25cl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to E\in B$$
44	32 17 18 16 41 19 42 43	syl322anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to E\in B$$
45		simp333	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
46	38 13	riotasv	$$⊢\left(E\in B\wedge z\in A\wedge \left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to E=O$$
47	44 26 27 45 46	syl112anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to E=O$$
48	47	oveq1d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to E\underset{\dot{}}{\vee }V=O\underset{\dot{}}{\vee }V$$
49	31 40 48	3brtr4d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(y\in A\wedge \neg y\underset{\dot{}}{\le }W\wedge \neg y\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to I\underset{\dot{}}{\le }\left(E\underset{\dot{}}{\vee }V\right)$$