Theorem dalaw

Description: Desargues's law, derived from Desargues's theorem dath and with no conditions on the atoms. If triples <. P , Q , R >. and <. S , T , U >. are centrally perspective, i.e. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) , then they are axially perspective. Theorem 13.3 of Crawley p. 110. (Contributed by NM, 7-Oct-2012)

	Ref	Expression
Hypotheses	dalaw.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalaw.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalaw.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalaw.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	dalaw	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dalaw.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		dalaw.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		dalaw.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		dalaw.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		eqid	$$⊢\mathrm{LPlanes}\left(K\right)=\mathrm{LPlanes}\left(K\right)$$
6	1 2 3 4 5	dalawlem14	$$⊢\left(\left(K\in \mathrm{HL}\wedge \neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$
7	6	3expib	$$⊢\left(K\in \mathrm{HL}\wedge \neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
8	7	3exp	$$⊢K\in \mathrm{HL}\to \left(\neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
9	1 2 3 4 5	dalawlem15	$$⊢\left(\left(K\in \mathrm{HL}\wedge \neg \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$
10	9	3expib	$$⊢\left(K\in \mathrm{HL}\wedge \neg \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
11	10	3exp	$$⊢K\in \mathrm{HL}\to \left(\neg \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
12		simp11	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to K\in \mathrm{HL}$$
13		simp2	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(P\in A\wedge Q\in A\wedge R\in A\right)$$
14		simp3	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(S\in A\wedge T\in A\wedge U\in A\right)$$
15		simp2ll	$$⊢\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)$$
16	15	3ad2ant1	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)$$
17		simp2rl	$$⊢\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)$$
18	17	3ad2ant1	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)$$
19		simp2lr	$$⊢\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
20	19	3ad2ant1	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
21		simp2rr	$$⊢\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)$$
22	21	3ad2ant1	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)$$
23		simp13	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)$$
24	1 2 3 4 5	dalawlem1	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$
25	12 13 14 16 18 20 22 23 24	syl323anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$
26	25	3expib	$$⊢\left(K\in \mathrm{HL}\wedge \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
27	26	3exp	$$⊢K\in \mathrm{HL}\to \left(\left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in \mathrm{LPlanes}\left(K\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
28	8 11 27	ecased	$$⊢K\in \mathrm{HL}\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)$$
29	28	exp4a	$$⊢K\in \mathrm{HL}\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\to \left(\left(S\in A\wedge T\in A\wedge U\in A\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
30	29	com34	$$⊢K\in \mathrm{HL}\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(S\in A\wedge T\in A\wedge U\in A\right)\to \left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
31	30	com24	$$⊢K\in \mathrm{HL}\to \left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\to \left(\left(S\in A\wedge T\in A\wedge U\in A\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
32	31	3imp	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$