Theorem cdleme21i

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

	Ref	Expression
Hypotheses	cdleme21.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme21.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme21.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme21.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme21.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme21.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme21.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme21g.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme21g.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme21g.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
	cdleme21g.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }D\right)$$
	cdleme21g.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Y\right)$$
Assertion	cdleme21i	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\to N=O\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme21.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme21.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme21.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme21.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme21.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme21.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme21.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		cdleme21g.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		cdleme21g.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		cdleme21g.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
11		cdleme21g.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }D\right)$$
12		cdleme21g.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Y\right)$$
13		simpl11	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\wedge \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
14		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
15		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
16		simp21l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to S\in A$$
17	14 15 16	3jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge S\in A\right)$$
18	17	adantr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\wedge \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\to \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge S\in A\right)$$
19		simp231	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to P\ne Q$$
20	19	adantr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\wedge \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\to P\ne Q$$
21		simp232	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
22	21	adantr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\wedge \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
23		simpr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\wedge \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\to \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)$$
24	1 2 4 5	4atexlem7	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$
25	13 18 20 22 23 24	syl113anc	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\wedge \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$
26	25	ex	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)$$
27	1 2 3 4 5 6 7 8 9 10 11 12	cdleme21h	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\to N=O\right)$$
28	26 27	syld	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\right)\to \left(\exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\to N=O\right)$$