Let x0 of type ι → ι → CT2 ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Claim L4:
(λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (80242.. x5) (b3303.. ((λ x6 . λ x7 : ι → ι → ι . λ x8 . λ x9 : ι → ι . x0 x6 x8 (λ x10 x11 . If_i (x10 = x6) (x9 x11) (x7 x10 x11))) x3 x4) x5) 4a7ef..) x1 (fb712.. (λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (80242.. x5) (b3303.. ((λ x6 . λ x7 : ι → ι → ι . λ x8 . λ x9 : ι → ι . x0 x6 x8 (λ x10 x11 . If_i (x10 = x6) (x9 x11) (x7 x10 x11))) x3 x4) x5) 4a7ef..)) x2 = x0 x1 x2 (ad3ed.. x0)Apply If_i_1 with
80242.. x2,
b3303.. ((λ x3 . λ x4 : ι → ι → ι . λ x5 . λ x6 : ι → ι . x0 x3 x5 (λ x7 x8 . If_i (x7 = x3) (x6 x8) (x4 x7 x8))) x1 (fb712.. (λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (80242.. x5) (b3303.. ((λ x6 . λ x7 : ι → ι → ι . λ x8 . λ x9 : ι → ι . x0 x6 x8 (λ x10 x11 . If_i (x10 = x6) (x9 x11) (x7 x10 x11))) x3 x4) x5) 4a7ef..))) x2,
4a7ef..,
λ x3 x4 . x4 = x0 x1 x2 (ad3ed.. x0) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_24de179dcb5553c68d4b421cdfd12754a4c26d470269d71da298b4a100368849 with
x0,
x1,
fb712.. (λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (80242.. x5) (b3303.. ((λ x6 . λ x7 : ι → ι → ι . λ x8 . λ x9 : ι → ι . x0 x6 x8 (λ x10 x11 . If_i (x10 = x6) (x9 x11) (x7 x10 x11))) x3 x4) x5) 4a7ef..),
x2,
λ x3 x4 . x4 = x0 x1 x2 (fb712.. (λ x5 . λ x6 : ι → ι → ι . λ x7 . If_i (80242.. x7) ... ...)) leaving 4 subgoals.
Apply unknownprop_8108c8b6b229f501aa74a149bad0d2abb28e649bcc6b26c7dc248be87be7ae6d with
λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (80242.. x5) (b3303.. ((λ x6 . λ x7 : ι → ι → ι . λ x8 . λ x9 : ι → ι . x0 x6 x8 (λ x10 x11 . If_i (x10 = x6) (x9 x11) (x7 x10 x11))) x3 x4) x5) 4a7ef..,
x1,
λ x3 x4 : ι → ι . x4 x2 = x0 x1 x2 (ad3ed.. x0) leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H1.
The subproof is completed by applying L4.