Theorem mulsass

Description: Associative law for surreal multiplication. Part of theorem 7 of Conway p. 19. Much like the case for additive groups, this theorem together with mulscom , addsdi , mulsgt0 , and the addition theorems would make the surreals into an ordered ring except that they are a proper class. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Assertion	mulsass	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ·s 𝐵 ) ·s 𝐶 ) = ( 𝐴 ·s ( 𝐵 ·s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s 𝑦 ) = ( 𝑥𝑂 ·s 𝑦 ) )
2	1	oveq1d	⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑥 ·s 𝑦 ) ·s 𝑧 ) = ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) )
3		oveq1	⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s ( 𝑦 ·s 𝑧 ) ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) )
4	2 3	eqeq12d	⊢ ( 𝑥 = 𝑥𝑂 → ( ( ( 𝑥 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧 ) ) ↔ ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ) )
5		oveq2	⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥𝑂 ·s 𝑦 ) = ( 𝑥𝑂 ·s 𝑦𝑂 ) )
6	5	oveq1d	⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) )
7		oveq1	⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 ·s 𝑧 ) = ( 𝑦𝑂 ·s 𝑧 ) )
8	7	oveq2d	⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) )
9	6 8	eqeq12d	⊢ ( 𝑦 = 𝑦𝑂 → ( ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ↔ ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) )
10		oveq2	⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) )
11		oveq2	⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑦𝑂 ·s 𝑧 ) = ( 𝑦𝑂 ·s 𝑧𝑂 ) )
12	11	oveq2d	⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) )
13	10 12	eqeq12d	⊢ ( 𝑧 = 𝑧𝑂 → ( ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ↔ ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ) )
14		oveq1	⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s 𝑦𝑂 ) = ( 𝑥𝑂 ·s 𝑦𝑂 ) )
15	14	oveq1d	⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) )
16		oveq1	⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) )
17	15 16	eqeq12d	⊢ ( 𝑥 = 𝑥𝑂 → ( ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ↔ ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ) )
18		oveq2	⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥 ·s 𝑦 ) = ( 𝑥 ·s 𝑦𝑂 ) )
19	18	oveq1d	⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) )
20		oveq1	⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 ·s 𝑧𝑂 ) = ( 𝑦𝑂 ·s 𝑧𝑂 ) )
21	20	oveq2d	⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) )
22	19 21	eqeq12d	⊢ ( 𝑦 = 𝑦𝑂 → ( ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ↔ ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ) )
23	5	oveq1d	⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) )
24	20	oveq2d	⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) )
25	23 24	eqeq12d	⊢ ( 𝑦 = 𝑦𝑂 → ( ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ↔ ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ) )
26		oveq2	⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) )
27	11	oveq2d	⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) )
28	26 27	eqeq12d	⊢ ( 𝑧 = 𝑧𝑂 → ( ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ↔ ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ) )
29		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·s 𝑦 ) = ( 𝐴 ·s 𝑦 ) )
30	29	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ·s 𝑦 ) ·s 𝑧 ) = ( ( 𝐴 ·s 𝑦 ) ·s 𝑧 ) )
31		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·s ( 𝑦 ·s 𝑧 ) ) = ( 𝐴 ·s ( 𝑦 ·s 𝑧 ) ) )
32	30 31	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧 ) ) ↔ ( ( 𝐴 ·s 𝑦 ) ·s 𝑧 ) = ( 𝐴 ·s ( 𝑦 ·s 𝑧 ) ) ) )
33		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 ·s 𝑦 ) = ( 𝐴 ·s 𝐵 ) )
34	33	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ·s 𝑦 ) ·s 𝑧 ) = ( ( 𝐴 ·s 𝐵 ) ·s 𝑧 ) )
35		oveq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 ·s 𝑧 ) = ( 𝐵 ·s 𝑧 ) )
36	35	oveq2d	⊢ ( 𝑦 = 𝐵 → ( 𝐴 ·s ( 𝑦 ·s 𝑧 ) ) = ( 𝐴 ·s ( 𝐵 ·s 𝑧 ) ) )
37	34 36	eqeq12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 ·s 𝑦 ) ·s 𝑧 ) = ( 𝐴 ·s ( 𝑦 ·s 𝑧 ) ) ↔ ( ( 𝐴 ·s 𝐵 ) ·s 𝑧 ) = ( 𝐴 ·s ( 𝐵 ·s 𝑧 ) ) ) )
38		oveq2	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 ·s 𝐵 ) ·s 𝑧 ) = ( ( 𝐴 ·s 𝐵 ) ·s 𝐶 ) )
39		oveq2	⊢ ( 𝑧 = 𝐶 → ( 𝐵 ·s 𝑧 ) = ( 𝐵 ·s 𝐶 ) )
40	39	oveq2d	⊢ ( 𝑧 = 𝐶 → ( 𝐴 ·s ( 𝐵 ·s 𝑧 ) ) = ( 𝐴 ·s ( 𝐵 ·s 𝐶 ) ) )
41	38 40	eqeq12d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ·s 𝐵 ) ·s 𝑧 ) = ( 𝐴 ·s ( 𝐵 ·s 𝑧 ) ) ↔ ( ( 𝐴 ·s 𝐵 ) ·s 𝐶 ) = ( 𝐴 ·s ( 𝐵 ·s 𝐶 ) ) ) )
42		simpl1	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → 𝑥 ∈ No )
43		simpl2	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → 𝑦 ∈ No )
44		simpl3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → 𝑧 ∈ No )
45		ssun1	⊢ ( L ‘ 𝑥 ) ⊆ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) )
46		ssun1	⊢ ( L ‘ 𝑦 ) ⊆ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) )
47		ssun1	⊢ ( L ‘ 𝑧 ) ⊆ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) )
48		simpr11	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) )
49		simpr12	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) )
50		simpr13	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) )
51		simpr22	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) )
52		simpr21	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) )
53		simpr23	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) )
54		simpr3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) )
55	42 43 44 45 46 47 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) ) )
56	55	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } )
57		ssun2	⊢ ( R ‘ 𝑥 ) ⊆ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) )
58		ssun2	⊢ ( R ‘ 𝑦 ) ⊆ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) )
59	42 43 44 57 58 47 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) ) )
60	59	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } )
61	56 60	uneq12d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) )
62		ssun2	⊢ ( R ‘ 𝑧 ) ⊆ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) )
63	42 43 44 45 58 62 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) ) )
64	63	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } )
65	42 43 44 57 46 62 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) ) )
66	65	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } )
67	64 66	uneq12d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ) )
68	61 67	uneq12d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ) ) )
69		un4	⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ) )
70		uncom	⊢ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } )
71	70	uneq2i	⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) )
72	69 71	eqtri	⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) )
73	68 72	eqtrdi	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ) )
74	42 43 44 45 46 62 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) ) )
75	74	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } )
76	42 43 44 57 58 62 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) ) )
77	76	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } )
78	75 77	uneq12d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) )
79	42 43 44 45 58 47 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) ) )
80	79	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } = { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } )
81	42 43 44 57 46 47 48 49 50 51 52 53 54	mulsasslem3	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) ) )
82	81	abbidv	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } = { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } )
83	80 82	uneq12d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ) )
84	78 83	uneq12d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ) ) )
85		un4	⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ) )
86		uncom	⊢ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ) = ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } )
87	86	uneq2i	⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) )
88	85 87	eqtri	⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) )
89	84 88	eqtrdi	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ) )
90	73 89	oveq12d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ) ) ) = ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ) ) )
91	42 43 44	mulsasslem1	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ( 𝑥 ·s 𝑦 ) ·s 𝑧 ) = ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ·s 𝑧𝑅 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑅 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ·s 𝑧𝑅 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝐿 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ·s 𝑧𝐿 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧 ) +s ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝐿 ) ) -s ( ( ( ( 𝑥𝑅 ·s 𝑦 ) +s ( 𝑥 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ·s 𝑧𝐿 ) ) } ) ) ) )
92	42 43 44	mulsasslem2	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( 𝑥 ·s ( 𝑦 ·s 𝑧 ) ) = ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝐿 ·s 𝑧𝑅 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝐿 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝐿 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝑅 ·s 𝑧𝐿 ) ) ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝐿 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝐿 ) ) -s ( 𝑦𝐿 ·s 𝑧𝐿 ) ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑎 = ( ( ( 𝑥𝑅 ·s ( 𝑦 ·s 𝑧 ) ) +s ( 𝑥 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) -s ( 𝑥𝑅 ·s ( ( ( 𝑦𝑅 ·s 𝑧 ) +s ( 𝑦 ·s 𝑧𝑅 ) ) -s ( 𝑦𝑅 ·s 𝑧𝑅 ) ) ) ) } ) ) ) )
93	90 91 92	3eqtr4d	⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ) → ( ( 𝑥 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧 ) ) )
94	93	ex	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥𝑂 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑥𝑂 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥𝑂 ·s ( 𝑦 ·s 𝑧 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧𝑂 ) ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑥 ·s 𝑦𝑂 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦𝑂 ·s 𝑧 ) ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑥 ·s 𝑦 ) ·s 𝑧𝑂 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧𝑂 ) ) ) → ( ( 𝑥 ·s 𝑦 ) ·s 𝑧 ) = ( 𝑥 ·s ( 𝑦 ·s 𝑧 ) ) ) )
95	4 9 13 17 22 25 28 32 37 41 94	no3inds	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ·s 𝐵 ) ·s 𝐶 ) = ( 𝐴 ·s ( 𝐵 ·s 𝐶 ) ) )