Theorem mulsval

Description: The value of surreal multiplication. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	mulsval	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ·s 𝐵 ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-muls	⊢ ·s = norec2 ( ( 𝑧 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) ) )
2	1	norec2ov	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ·s 𝐵 ) = ( ⟨ 𝐴 , 𝐵 ⟩ ( 𝑧 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) ) ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ) )
3		opex	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ V
4		mulsfn	⊢ ·s Fn ( No × No )
5		fnfun	⊢ ( ·s Fn ( No × No ) → Fun ·s )
6	4 5	ax-mp	⊢ Fun ·s
7		fvex	⊢ ( L ‘ 𝐴 ) ∈ V
8		fvex	⊢ ( R ‘ 𝐴 ) ∈ V
9	7 8	unex	⊢ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∈ V
10		snex	⊢ { 𝐴 } ∈ V
11	9 10	unex	⊢ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) ∈ V
12		fvex	⊢ ( L ‘ 𝐵 ) ∈ V
13		fvex	⊢ ( R ‘ 𝐵 ) ∈ V
14	12 13	unex	⊢ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∈ V
15		snex	⊢ { 𝐵 } ∈ V
16	14 15	unex	⊢ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ∈ V
17	11 16	xpex	⊢ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∈ V
18	17	difexi	⊢ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ∈ V
19		resfunexg	⊢ ( ( Fun ·s ∧ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ∈ V ) → ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ∈ V )
20	6 18 19	mp2an	⊢ ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ∈ V
21		fveq2	⊢ ( 𝑧 = ⟨ 𝐴 , 𝐵 ⟩ → ( 1st ‘ 𝑧 ) = ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
22		fveq2	⊢ ( 𝑧 = ⟨ 𝐴 , 𝐵 ⟩ → ( 2nd ‘ 𝑧 ) = ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
23	22	csbeq1d	⊢ ( 𝑧 = ⟨ 𝐴 , 𝐵 ⟩ → ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) = ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) )
24	21 23	csbeq12dv	⊢ ( 𝑧 = ⟨ 𝐴 , 𝐵 ⟩ → ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) = ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) )
25		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑝 𝑚 𝑦 ) = ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) )
26		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑥 𝑚 𝑞 ) = ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) )
27	25 26	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) = ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) )
28		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑝 𝑚 𝑞 ) = ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) )
29	27 28	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) )
30	29	eqeq2d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) ↔ 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
31	30	2rexbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) ↔ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
32	31	abbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } = { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } )
33		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑟 𝑚 𝑦 ) = ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) )
34		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑥 𝑚 𝑠 ) = ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) )
35	33 34	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) = ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) )
36		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑟 𝑚 𝑠 ) = ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) )
37	35 36	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) )
38	37	eqeq2d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) ↔ 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
39	38	2rexbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) ↔ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
40	39	abbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } = { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } )
41	32 40	uneq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) )
42		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑡 𝑚 𝑦 ) = ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) )
43		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑥 𝑚 𝑢 ) = ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) )
44	42 43	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) = ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) )
45		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑡 𝑚 𝑢 ) = ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) )
46	44 45	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) )
47	46	eqeq2d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) ↔ 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
48	47	2rexbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) ↔ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
49	48	abbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } = { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } )
50		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑣 𝑚 𝑦 ) = ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) )
51		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑥 𝑚 𝑤 ) = ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) )
52	50 51	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) = ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) )
53		oveq	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑣 𝑚 𝑤 ) = ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) )
54	52 53	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) )
55	54	eqeq2d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) ↔ 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
56	55	2rexbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) ↔ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
57	56	abbidv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } = { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } )
58	49 57	uneq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) = ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) )
59	41 58	oveq12d	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
60	59	csbeq2dv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) = ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
61	60	csbeq2dv	⊢ ( 𝑚 = ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) → ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) = ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
62		eqid	⊢ ( 𝑧 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) ) = ( 𝑧 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) )
63		ovex	⊢ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) ∈ V
64	63	csbex	⊢ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) ∈ V
65	64	csbex	⊢ ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) ∈ V
66	24 61 62 65	ovmpo	⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ ∈ V ∧ ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ ( 𝑧 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) ) ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ) = ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
67	3 20 66	mp2an	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ( 𝑧 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) ) ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ) = ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) )
68		op1stg	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐴 )
69	68	csbeq1d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ⦋ 𝐴 / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
70		op2ndg	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐵 )
71	70	csbeq1d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
72	71	csbeq2dv	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ⦋ 𝐴 / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
73		simpl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
74		fveq2	⊢ ( 𝑥 = 𝐴 → ( L ‘ 𝑥 ) = ( L ‘ 𝐴 ) )
75		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) = ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) )
76	75	oveq2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) = ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) )
77	76	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) )
78	77	eqeq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
79	78	rexbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
80	74 79	rexeqbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
81	80	abbidv	⊢ ( 𝑥 = 𝐴 → { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } = { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } )
82		fveq2	⊢ ( 𝑥 = 𝐴 → ( R ‘ 𝑥 ) = ( R ‘ 𝐴 ) )
83		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) = ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) )
84	83	oveq2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) = ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) )
85	84	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) )
86	85	eqeq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
87	86	rexbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
88	82 87	rexeqbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
89	88	abbidv	⊢ ( 𝑥 = 𝐴 → { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } = { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } )
90	81 89	uneq12d	⊢ ( 𝑥 = 𝐴 → ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) )
91		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) = ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) )
92	91	oveq2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) = ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) )
93	92	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) )
94	93	eqeq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
95	94	rexbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
96	74 95	rexeqbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
97	96	abbidv	⊢ ( 𝑥 = 𝐴 → { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } = { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } )
98		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) = ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) )
99	98	oveq2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) = ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) )
100	99	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) )
101	100	eqeq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
102	101	rexbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
103	82 102	rexeqbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
104	103	abbidv	⊢ ( 𝑥 = 𝐴 → { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } = { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } )
105	97 104	uneq12d	⊢ ( 𝑥 = 𝐴 → ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) = ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) )
106	90 105	oveq12d	⊢ ( 𝑥 = 𝐴 → ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
107	106	csbeq2dv	⊢ ( 𝑥 = 𝐴 → ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
108	107	adantl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑥 = 𝐴 ) → ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
109	73 108	csbied	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
110		simpr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
111		fveq2	⊢ ( 𝑦 = 𝐵 → ( L ‘ 𝑦 ) = ( L ‘ 𝐵 ) )
112		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) = ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) )
113	112	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) = ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) )
114	113	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) )
115	114	eqeq2d	⊢ ( 𝑦 = 𝐵 → ( 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
116	111 115	rexeqbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
117	116	rexbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ) )
118	117	abbidv	⊢ ( 𝑦 = 𝐵 → { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } = { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } )
119		fveq2	⊢ ( 𝑦 = 𝐵 → ( R ‘ 𝑦 ) = ( R ‘ 𝐵 ) )
120		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) = ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) )
121	120	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) = ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) )
122	121	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) )
123	122	eqeq2d	⊢ ( 𝑦 = 𝐵 → ( 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
124	119 123	rexeqbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
125	124	rexbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ) )
126	125	abbidv	⊢ ( 𝑦 = 𝐵 → { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } = { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } )
127	118 126	uneq12d	⊢ ( 𝑦 = 𝐵 → ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) )
128		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) = ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) )
129	128	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) = ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) )
130	129	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) )
131	130	eqeq2d	⊢ ( 𝑦 = 𝐵 → ( 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
132	119 131	rexeqbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
133	132	rexbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ) )
134	133	abbidv	⊢ ( 𝑦 = 𝐵 → { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } = { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } )
135		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) = ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) )
136	135	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) = ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) )
137	136	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) )
138	137	eqeq2d	⊢ ( 𝑦 = 𝐵 → ( 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
139	111 138	rexeqbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
140	139	rexbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ) )
141	140	abbidv	⊢ ( 𝑦 = 𝐵 → { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } = { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } )
142	134 141	uneq12d	⊢ ( 𝑦 = 𝐵 → ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) = ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) )
143	127 142	oveq12d	⊢ ( 𝑦 = 𝐵 → ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
144	143	adantl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑦 = 𝐵 ) → ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
145	110 144	csbied	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) )
146		elun1	⊢ ( 𝑝 ∈ ( L ‘ 𝐴 ) → 𝑝 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
147	146	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝑝 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
148		elun1	⊢ ( 𝑝 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) → 𝑝 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
149	147 148	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝑝 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
150		snidg	⊢ ( 𝐵 ∈ No → 𝐵 ∈ { 𝐵 } )
151	150	adantl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ { 𝐵 } )
152		elun2	⊢ ( 𝐵 ∈ { 𝐵 } → 𝐵 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
153	151 152	syl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
154	153	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝐵 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
155	149 154	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑝 , 𝐵 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
156		leftirr	⊢ ¬ 𝐴 ∈ ( L ‘ 𝐴 )
157		eleq1	⊢ ( 𝑝 = 𝐴 → ( 𝑝 ∈ ( L ‘ 𝐴 ) ↔ 𝐴 ∈ ( L ‘ 𝐴 ) ) )
158	156 157	mtbiri	⊢ ( 𝑝 = 𝐴 → ¬ 𝑝 ∈ ( L ‘ 𝐴 ) )
159	158	necon2ai	⊢ ( 𝑝 ∈ ( L ‘ 𝐴 ) → 𝑝 ≠ 𝐴 )
160	159	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝑝 ≠ 𝐴 )
161	160	orcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) )
162		vex	⊢ 𝑝 ∈ V
163		opthneg	⊢ ( ( 𝑝 ∈ V ∧ 𝐵 ∈ No ) → ( ⟨ 𝑝 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
164	162 163	mpan	⊢ ( 𝐵 ∈ No → ( ⟨ 𝑝 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
165	164	ad2antlr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( ⟨ 𝑝 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
166	161 165	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑝 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
167		opex	⊢ ⟨ 𝑝 , 𝐵 ⟩ ∈ V
168	167	elsn	⊢ ( ⟨ 𝑝 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑝 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
169	168	necon3bbii	⊢ ( ¬ ⟨ 𝑝 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑝 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
170	166 169	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑝 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
171	155 170	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑝 , 𝐵 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
172	171	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑝 , 𝐵 ⟩ ) = ( ·s ‘ ⟨ 𝑝 , 𝐵 ⟩ ) )
173		df-ov	⊢ ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑝 , 𝐵 ⟩ )
174		df-ov	⊢ ( 𝑝 ·s 𝐵 ) = ( ·s ‘ ⟨ 𝑝 , 𝐵 ⟩ )
175	172 173 174	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( 𝑝 ·s 𝐵 ) )
176		snidg	⊢ ( 𝐴 ∈ No → 𝐴 ∈ { 𝐴 } )
177	176	adantr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ { 𝐴 } )
178		elun2	⊢ ( 𝐴 ∈ { 𝐴 } → 𝐴 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
179	177 178	syl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
180	179	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝐴 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
181		elun1	⊢ ( 𝑞 ∈ ( L ‘ 𝐵 ) → 𝑞 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
182	181	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝑞 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
183		elun1	⊢ ( 𝑞 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) → 𝑞 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
184	182 183	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝑞 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
185	180 184	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑞 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
186		leftirr	⊢ ¬ 𝐵 ∈ ( L ‘ 𝐵 )
187		eleq1	⊢ ( 𝑞 = 𝐵 → ( 𝑞 ∈ ( L ‘ 𝐵 ) ↔ 𝐵 ∈ ( L ‘ 𝐵 ) ) )
188	186 187	mtbiri	⊢ ( 𝑞 = 𝐵 → ¬ 𝑞 ∈ ( L ‘ 𝐵 ) )
189	188	necon2ai	⊢ ( 𝑞 ∈ ( L ‘ 𝐵 ) → 𝑞 ≠ 𝐵 )
190	189	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → 𝑞 ≠ 𝐵 )
191	190	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵 ) )
192		opthneg	⊢ ( ( 𝐴 ∈ No ∧ 𝑞 ∈ V ) → ( ⟨ 𝐴 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵 ) ) )
193	192	elvd	⊢ ( 𝐴 ∈ No → ( ⟨ 𝐴 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵 ) ) )
194	193	ad2antrr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( ⟨ 𝐴 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵 ) ) )
195	191 194	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
196		opex	⊢ ⟨ 𝐴 , 𝑞 ⟩ ∈ V
197	196	elsn	⊢ ( ⟨ 𝐴 , 𝑞 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑞 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
198	197	necon3bbii	⊢ ( ¬ ⟨ 𝐴 , 𝑞 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
199	195 198	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ¬ ⟨ 𝐴 , 𝑞 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
200	185 199	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑞 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
201	200	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑞 ⟩ ) = ( ·s ‘ ⟨ 𝐴 , 𝑞 ⟩ ) )
202		df-ov	⊢ ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑞 ⟩ )
203		df-ov	⊢ ( 𝐴 ·s 𝑞 ) = ( ·s ‘ ⟨ 𝐴 , 𝑞 ⟩ )
204	201 202 203	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) = ( 𝐴 ·s 𝑞 ) )
205	175 204	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) = ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) )
206	149 184	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑝 , 𝑞 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
207	190	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑝 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵 ) )
208		vex	⊢ 𝑞 ∈ V
209	162 208	opthne	⊢ ( ⟨ 𝑝 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑝 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵 ) )
210	207 209	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑝 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
211		opex	⊢ ⟨ 𝑝 , 𝑞 ⟩ ∈ V
212	211	elsn	⊢ ( ⟨ 𝑝 , 𝑞 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑝 , 𝑞 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
213	212	necon3bbii	⊢ ( ¬ ⟨ 𝑝 , 𝑞 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑝 , 𝑞 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
214	210 213	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑝 , 𝑞 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
215	206 214	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑝 , 𝑞 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
216	215	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑝 , 𝑞 ⟩ ) = ( ·s ‘ ⟨ 𝑝 , 𝑞 ⟩ ) )
217		df-ov	⊢ ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑝 , 𝑞 ⟩ )
218		df-ov	⊢ ( 𝑝 ·s 𝑞 ) = ( ·s ‘ ⟨ 𝑝 , 𝑞 ⟩ )
219	216 217 218	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) = ( 𝑝 ·s 𝑞 ) )
220	205 219	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) )
221	220	eqeq2d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑝 ∈ ( L ‘ 𝐴 ) ∧ 𝑞 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) ) )
222	221	2rexbidva	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) ↔ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) ) )
223	222	abbidv	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } = { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } )
224		elun2	⊢ ( 𝑟 ∈ ( R ‘ 𝐴 ) → 𝑟 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
225	224	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝑟 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
226		elun1	⊢ ( 𝑟 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) → 𝑟 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
227	225 226	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝑟 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
228	153	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝐵 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
229	227 228	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑟 , 𝐵 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
230		rightirr	⊢ ¬ 𝐴 ∈ ( R ‘ 𝐴 )
231		eleq1	⊢ ( 𝑟 = 𝐴 → ( 𝑟 ∈ ( R ‘ 𝐴 ) ↔ 𝐴 ∈ ( R ‘ 𝐴 ) ) )
232	230 231	mtbiri	⊢ ( 𝑟 = 𝐴 → ¬ 𝑟 ∈ ( R ‘ 𝐴 ) )
233	232	necon2ai	⊢ ( 𝑟 ∈ ( R ‘ 𝐴 ) → 𝑟 ≠ 𝐴 )
234	233	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝑟 ≠ 𝐴 )
235	234	orcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) )
236		vex	⊢ 𝑟 ∈ V
237		opthneg	⊢ ( ( 𝑟 ∈ V ∧ 𝐵 ∈ No ) → ( ⟨ 𝑟 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
238	236 237	mpan	⊢ ( 𝐵 ∈ No → ( ⟨ 𝑟 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
239	238	ad2antlr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( ⟨ 𝑟 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
240	235 239	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑟 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
241		opex	⊢ ⟨ 𝑟 , 𝐵 ⟩ ∈ V
242	241	elsn	⊢ ( ⟨ 𝑟 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑟 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
243	242	necon3bbii	⊢ ( ¬ ⟨ 𝑟 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑟 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
244	240 243	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑟 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
245	229 244	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑟 , 𝐵 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
246	245	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑟 , 𝐵 ⟩ ) = ( ·s ‘ ⟨ 𝑟 , 𝐵 ⟩ ) )
247		df-ov	⊢ ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑟 , 𝐵 ⟩ )
248		df-ov	⊢ ( 𝑟 ·s 𝐵 ) = ( ·s ‘ ⟨ 𝑟 , 𝐵 ⟩ )
249	246 247 248	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( 𝑟 ·s 𝐵 ) )
250	179	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝐴 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
251		elun2	⊢ ( 𝑠 ∈ ( R ‘ 𝐵 ) → 𝑠 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
252	251	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝑠 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
253		elun1	⊢ ( 𝑠 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) → 𝑠 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
254	252 253	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝑠 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
255	250 254	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑠 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
256		rightirr	⊢ ¬ 𝐵 ∈ ( R ‘ 𝐵 )
257		eleq1	⊢ ( 𝑠 = 𝐵 → ( 𝑠 ∈ ( R ‘ 𝐵 ) ↔ 𝐵 ∈ ( R ‘ 𝐵 ) ) )
258	256 257	mtbiri	⊢ ( 𝑠 = 𝐵 → ¬ 𝑠 ∈ ( R ‘ 𝐵 ) )
259	258	necon2ai	⊢ ( 𝑠 ∈ ( R ‘ 𝐵 ) → 𝑠 ≠ 𝐵 )
260	259	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → 𝑠 ≠ 𝐵 )
261	260	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵 ) )
262		opthneg	⊢ ( ( 𝐴 ∈ No ∧ 𝑠 ∈ V ) → ( ⟨ 𝐴 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵 ) ) )
263	262	elvd	⊢ ( 𝐴 ∈ No → ( ⟨ 𝐴 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵 ) ) )
264	263	ad2antrr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( ⟨ 𝐴 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵 ) ) )
265	261 264	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
266		opex	⊢ ⟨ 𝐴 , 𝑠 ⟩ ∈ V
267	266	elsn	⊢ ( ⟨ 𝐴 , 𝑠 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑠 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
268	267	necon3bbii	⊢ ( ¬ ⟨ 𝐴 , 𝑠 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
269	265 268	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ¬ ⟨ 𝐴 , 𝑠 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
270	255 269	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑠 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
271	270	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑠 ⟩ ) = ( ·s ‘ ⟨ 𝐴 , 𝑠 ⟩ ) )
272		df-ov	⊢ ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑠 ⟩ )
273		df-ov	⊢ ( 𝐴 ·s 𝑠 ) = ( ·s ‘ ⟨ 𝐴 , 𝑠 ⟩ )
274	271 272 273	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) = ( 𝐴 ·s 𝑠 ) )
275	249 274	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) = ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) )
276	227 254	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑟 , 𝑠 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
277	260	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑟 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵 ) )
278		vex	⊢ 𝑠 ∈ V
279	236 278	opthne	⊢ ( ⟨ 𝑟 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑟 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵 ) )
280	277 279	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑟 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
281		opex	⊢ ⟨ 𝑟 , 𝑠 ⟩ ∈ V
282	281	elsn	⊢ ( ⟨ 𝑟 , 𝑠 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑟 , 𝑠 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
283	282	necon3bbii	⊢ ( ¬ ⟨ 𝑟 , 𝑠 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑟 , 𝑠 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
284	280 283	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑟 , 𝑠 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
285	276 284	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑟 , 𝑠 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
286	285	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑟 , 𝑠 ⟩ ) = ( ·s ‘ ⟨ 𝑟 , 𝑠 ⟩ ) )
287		df-ov	⊢ ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑟 , 𝑠 ⟩ )
288		df-ov	⊢ ( 𝑟 ·s 𝑠 ) = ( ·s ‘ ⟨ 𝑟 , 𝑠 ⟩ )
289	286 287 288	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) = ( 𝑟 ·s 𝑠 ) )
290	275 289	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) )
291	290	eqeq2d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑟 ∈ ( R ‘ 𝐴 ) ∧ 𝑠 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) ) )
292	291	2rexbidva	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) ↔ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) ) )
293	292	abbidv	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } = { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } )
294	223 293	uneq12d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) = ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } ) )
295		elun1	⊢ ( 𝑡 ∈ ( L ‘ 𝐴 ) → 𝑡 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
296	295	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝑡 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
297		elun1	⊢ ( 𝑡 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) → 𝑡 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
298	296 297	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝑡 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
299	153	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝐵 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
300	298 299	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑡 , 𝐵 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
301		eleq1	⊢ ( 𝑡 = 𝐴 → ( 𝑡 ∈ ( L ‘ 𝐴 ) ↔ 𝐴 ∈ ( L ‘ 𝐴 ) ) )
302	156 301	mtbiri	⊢ ( 𝑡 = 𝐴 → ¬ 𝑡 ∈ ( L ‘ 𝐴 ) )
303	302	necon2ai	⊢ ( 𝑡 ∈ ( L ‘ 𝐴 ) → 𝑡 ≠ 𝐴 )
304	303	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝑡 ≠ 𝐴 )
305	304	orcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) )
306		vex	⊢ 𝑡 ∈ V
307		opthneg	⊢ ( ( 𝑡 ∈ V ∧ 𝐵 ∈ No ) → ( ⟨ 𝑡 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
308	306 307	mpan	⊢ ( 𝐵 ∈ No → ( ⟨ 𝑡 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
309	308	ad2antlr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( ⟨ 𝑡 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
310	305 309	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑡 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
311		opex	⊢ ⟨ 𝑡 , 𝐵 ⟩ ∈ V
312	311	elsn	⊢ ( ⟨ 𝑡 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑡 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
313	312	necon3bbii	⊢ ( ¬ ⟨ 𝑡 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑡 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
314	310 313	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑡 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
315	300 314	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑡 , 𝐵 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
316	315	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑡 , 𝐵 ⟩ ) = ( ·s ‘ ⟨ 𝑡 , 𝐵 ⟩ ) )
317		df-ov	⊢ ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑡 , 𝐵 ⟩ )
318		df-ov	⊢ ( 𝑡 ·s 𝐵 ) = ( ·s ‘ ⟨ 𝑡 , 𝐵 ⟩ )
319	316 317 318	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( 𝑡 ·s 𝐵 ) )
320	179	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝐴 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
321		elun2	⊢ ( 𝑢 ∈ ( R ‘ 𝐵 ) → 𝑢 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
322	321	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝑢 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
323		elun1	⊢ ( 𝑢 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) → 𝑢 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
324	322 323	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝑢 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
325	320 324	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑢 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
326		eleq1	⊢ ( 𝑢 = 𝐵 → ( 𝑢 ∈ ( R ‘ 𝐵 ) ↔ 𝐵 ∈ ( R ‘ 𝐵 ) ) )
327	256 326	mtbiri	⊢ ( 𝑢 = 𝐵 → ¬ 𝑢 ∈ ( R ‘ 𝐵 ) )
328	327	necon2ai	⊢ ( 𝑢 ∈ ( R ‘ 𝐵 ) → 𝑢 ≠ 𝐵 )
329	328	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → 𝑢 ≠ 𝐵 )
330	329	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵 ) )
331		opthneg	⊢ ( ( 𝐴 ∈ No ∧ 𝑢 ∈ V ) → ( ⟨ 𝐴 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵 ) ) )
332	331	elvd	⊢ ( 𝐴 ∈ No → ( ⟨ 𝐴 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵 ) ) )
333	332	ad2antrr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( ⟨ 𝐴 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵 ) ) )
334	330 333	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
335		opex	⊢ ⟨ 𝐴 , 𝑢 ⟩ ∈ V
336	335	elsn	⊢ ( ⟨ 𝐴 , 𝑢 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑢 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
337	336	necon3bbii	⊢ ( ¬ ⟨ 𝐴 , 𝑢 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
338	334 337	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ¬ ⟨ 𝐴 , 𝑢 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
339	325 338	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑢 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
340	339	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑢 ⟩ ) = ( ·s ‘ ⟨ 𝐴 , 𝑢 ⟩ ) )
341		df-ov	⊢ ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑢 ⟩ )
342		df-ov	⊢ ( 𝐴 ·s 𝑢 ) = ( ·s ‘ ⟨ 𝐴 , 𝑢 ⟩ )
343	340 341 342	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) = ( 𝐴 ·s 𝑢 ) )
344	319 343	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) = ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) )
345	298 324	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑡 , 𝑢 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
346	329	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑡 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵 ) )
347		vex	⊢ 𝑢 ∈ V
348	306 347	opthne	⊢ ( ⟨ 𝑡 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑡 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵 ) )
349	346 348	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑡 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
350		opex	⊢ ⟨ 𝑡 , 𝑢 ⟩ ∈ V
351	350	elsn	⊢ ( ⟨ 𝑡 , 𝑢 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑡 , 𝑢 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
352	351	necon3bbii	⊢ ( ¬ ⟨ 𝑡 , 𝑢 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑡 , 𝑢 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
353	349 352	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑡 , 𝑢 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
354	345 353	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ⟨ 𝑡 , 𝑢 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
355	354	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑡 , 𝑢 ⟩ ) = ( ·s ‘ ⟨ 𝑡 , 𝑢 ⟩ ) )
356		df-ov	⊢ ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑡 , 𝑢 ⟩ )
357		df-ov	⊢ ( 𝑡 ·s 𝑢 ) = ( ·s ‘ ⟨ 𝑡 , 𝑢 ⟩ )
358	355 356 357	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) = ( 𝑡 ·s 𝑢 ) )
359	344 358	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) )
360	359	eqeq2d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑡 ∈ ( L ‘ 𝐴 ) ∧ 𝑢 ∈ ( R ‘ 𝐵 ) ) ) → ( 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) ) )
361	360	2rexbidva	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) ↔ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) ) )
362	361	abbidv	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } = { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } )
363		elun2	⊢ ( 𝑣 ∈ ( R ‘ 𝐴 ) → 𝑣 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
364	363	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝑣 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) )
365		elun1	⊢ ( 𝑣 ∈ ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) → 𝑣 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
366	364 365	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝑣 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
367	153	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝐵 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
368	366 367	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑣 , 𝐵 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
369		eleq1	⊢ ( 𝑣 = 𝐴 → ( 𝑣 ∈ ( R ‘ 𝐴 ) ↔ 𝐴 ∈ ( R ‘ 𝐴 ) ) )
370	230 369	mtbiri	⊢ ( 𝑣 = 𝐴 → ¬ 𝑣 ∈ ( R ‘ 𝐴 ) )
371	370	necon2ai	⊢ ( 𝑣 ∈ ( R ‘ 𝐴 ) → 𝑣 ≠ 𝐴 )
372	371	ad2antrl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝑣 ≠ 𝐴 )
373	372	orcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) )
374		vex	⊢ 𝑣 ∈ V
375		opthneg	⊢ ( ( 𝑣 ∈ V ∧ 𝐵 ∈ No ) → ( ⟨ 𝑣 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
376	374 375	mpan	⊢ ( 𝐵 ∈ No → ( ⟨ 𝑣 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
377	376	ad2antlr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( ⟨ 𝑣 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵 ) ) )
378	373 377	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑣 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
379		opex	⊢ ⟨ 𝑣 , 𝐵 ⟩ ∈ V
380	379	elsn	⊢ ( ⟨ 𝑣 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑣 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
381	380	necon3bbii	⊢ ( ¬ ⟨ 𝑣 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑣 , 𝐵 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
382	378 381	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑣 , 𝐵 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
383	368 382	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑣 , 𝐵 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
384	383	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑣 , 𝐵 ⟩ ) = ( ·s ‘ ⟨ 𝑣 , 𝐵 ⟩ ) )
385		df-ov	⊢ ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑣 , 𝐵 ⟩ )
386		df-ov	⊢ ( 𝑣 ·s 𝐵 ) = ( ·s ‘ ⟨ 𝑣 , 𝐵 ⟩ )
387	384 385 386	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) = ( 𝑣 ·s 𝐵 ) )
388	179	adantr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝐴 ∈ ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) )
389		elun1	⊢ ( 𝑤 ∈ ( L ‘ 𝐵 ) → 𝑤 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
390	389	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝑤 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) )
391		elun1	⊢ ( 𝑤 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) → 𝑤 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
392	390 391	syl	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝑤 ∈ ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) )
393	388 392	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑤 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
394		eleq1	⊢ ( 𝑤 = 𝐵 → ( 𝑤 ∈ ( L ‘ 𝐵 ) ↔ 𝐵 ∈ ( L ‘ 𝐵 ) ) )
395	186 394	mtbiri	⊢ ( 𝑤 = 𝐵 → ¬ 𝑤 ∈ ( L ‘ 𝐵 ) )
396	395	necon2ai	⊢ ( 𝑤 ∈ ( L ‘ 𝐵 ) → 𝑤 ≠ 𝐵 )
397	396	ad2antll	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → 𝑤 ≠ 𝐵 )
398	397	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵 ) )
399		opthneg	⊢ ( ( 𝐴 ∈ No ∧ 𝑤 ∈ V ) → ( ⟨ 𝐴 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵 ) ) )
400	399	elvd	⊢ ( 𝐴 ∈ No → ( ⟨ 𝐴 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵 ) ) )
401	400	ad2antrr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( ⟨ 𝐴 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵 ) ) )
402	398 401	mpbird	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
403		opex	⊢ ⟨ 𝐴 , 𝑤 ⟩ ∈ V
404	403	elsn	⊢ ( ⟨ 𝐴 , 𝑤 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑤 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
405	404	necon3bbii	⊢ ( ¬ ⟨ 𝐴 , 𝑤 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝐴 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
406	402 405	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ¬ ⟨ 𝐴 , 𝑤 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
407	393 406	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝐴 , 𝑤 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
408	407	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑤 ⟩ ) = ( ·s ‘ ⟨ 𝐴 , 𝑤 ⟩ ) )
409		df-ov	⊢ ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝐴 , 𝑤 ⟩ )
410		df-ov	⊢ ( 𝐴 ·s 𝑤 ) = ( ·s ‘ ⟨ 𝐴 , 𝑤 ⟩ )
411	408 409 410	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) = ( 𝐴 ·s 𝑤 ) )
412	387 411	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) = ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) )
413	366 392	opelxpd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑣 , 𝑤 ⟩ ∈ ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) )
414	397	olcd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑣 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵 ) )
415		vex	⊢ 𝑤 ∈ V
416	374 415	opthne	⊢ ( ⟨ 𝑣 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑣 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵 ) )
417	414 416	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑣 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
418		opex	⊢ ⟨ 𝑣 , 𝑤 ⟩ ∈ V
419	418	elsn	⊢ ( ⟨ 𝑣 , 𝑤 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑣 , 𝑤 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
420	419	necon3bbii	⊢ ( ¬ ⟨ 𝑣 , 𝑤 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } ↔ ⟨ 𝑣 , 𝑤 ⟩ ≠ ⟨ 𝐴 , 𝐵 ⟩ )
421	417 420	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ¬ ⟨ 𝑣 , 𝑤 ⟩ ∈ { ⟨ 𝐴 , 𝐵 ⟩ } )
422	413 421	eldifd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ⟨ 𝑣 , 𝑤 ⟩ ∈ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
423	422	fvresd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑣 , 𝑤 ⟩ ) = ( ·s ‘ ⟨ 𝑣 , 𝑤 ⟩ ) )
424		df-ov	⊢ ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) = ( ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ‘ ⟨ 𝑣 , 𝑤 ⟩ )
425		df-ov	⊢ ( 𝑣 ·s 𝑤 ) = ( ·s ‘ ⟨ 𝑣 , 𝑤 ⟩ )
426	423 424 425	3eqtr4g	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) = ( 𝑣 ·s 𝑤 ) )
427	412 426	oveq12d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) )
428	427	eqeq2d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑣 ∈ ( R ‘ 𝐴 ) ∧ 𝑤 ∈ ( L ‘ 𝐵 ) ) ) → ( 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) ) )
429	428	2rexbidva	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) ↔ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) ) )
430	429	abbidv	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } = { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } )
431	362 430	uneq12d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) = ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } ) )
432	294 431	oveq12d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝐵 ) +s ( 𝐴 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } ) ) )
433	109 145 432	3eqtrd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } ) ) )
434	69 72 433	3eqtrd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ⦋ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑥 ⦌ ⦋ ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) -s ( 𝑝 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) -s ( 𝑟 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) -s ( 𝑡 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑦 ) +s ( 𝑥 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) -s ( 𝑣 ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) 𝑤 ) ) } ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } ) ) )
435	67 434	eqtrid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ⟨ 𝐴 , 𝐵 ⟩ ( 𝑧 ∈ V , 𝑚 ∈ V ↦ ⦋ ( 1st ‘ 𝑧 ) / 𝑥 ⦌ ⦋ ( 2nd ‘ 𝑧 ) / 𝑦 ⦌ ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝑥 ) ∃ 𝑞 ∈ ( L ‘ 𝑦 ) 𝑎 = ( ( ( 𝑝 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑞 ) ) -s ( 𝑝 𝑚 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑥 ) ∃ 𝑠 ∈ ( R ‘ 𝑦 ) 𝑏 = ( ( ( 𝑟 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑠 ) ) -s ( 𝑟 𝑚 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝑥 ) ∃ 𝑢 ∈ ( R ‘ 𝑦 ) 𝑐 = ( ( ( 𝑡 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑢 ) ) -s ( 𝑡 𝑚 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝑥 ) ∃ 𝑤 ∈ ( L ‘ 𝑦 ) 𝑑 = ( ( ( 𝑣 𝑚 𝑦 ) +s ( 𝑥 𝑚 𝑤 ) ) -s ( 𝑣 𝑚 𝑤 ) ) } ) ) ) ( ·s ↾ ( ( ( ( ( L ‘ 𝐴 ) ∪ ( R ‘ 𝐴 ) ) ∪ { 𝐴 } ) × ( ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ∪ { 𝐵 } ) ) ∖ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } ) ) )
436	2 435	eqtrd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ·s 𝐵 ) = ( ( { 𝑎 ∣ ∃ 𝑝 ∈ ( L ‘ 𝐴 ) ∃ 𝑞 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( 𝑝 ·s 𝐵 ) +s ( 𝐴 ·s 𝑞 ) ) -s ( 𝑝 ·s 𝑞 ) ) } ∪ { 𝑏 ∣ ∃ 𝑟 ∈ ( R ‘ 𝐴 ) ∃ 𝑠 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑟 ·s 𝐵 ) +s ( 𝐴 ·s 𝑠 ) ) -s ( 𝑟 ·s 𝑠 ) ) } ) |s ( { 𝑐 ∣ ∃ 𝑡 ∈ ( L ‘ 𝐴 ) ∃ 𝑢 ∈ ( R ‘ 𝐵 ) 𝑐 = ( ( ( 𝑡 ·s 𝐵 ) +s ( 𝐴 ·s 𝑢 ) ) -s ( 𝑡 ·s 𝑢 ) ) } ∪ { 𝑑 ∣ ∃ 𝑣 ∈ ( R ‘ 𝐴 ) ∃ 𝑤 ∈ ( L ‘ 𝐵 ) 𝑑 = ( ( ( 𝑣 ·s 𝐵 ) +s ( 𝐴 ·s 𝑤 ) ) -s ( 𝑣 ·s 𝑤 ) ) } ) ) )