Assume H0: ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x3 ∈ x1 ⟶ ap (lam x1 (λ x4 . If_i (x4 = x3) x0 (x2 (ordsucc x3) x4))) x3 = x0.

Assume H1: ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 x4 . (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ ap (lam x1 (λ x5 . If_i (x5 = x3) x0 (x2 (ordsucc x3) x5))) x4 = ap (lam x1 (λ x5 . x2 (ordsucc x3) x5)) x4.

Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Let x3 of type ι be given.

Let x4 of type ι be given.

Let x5 of type ι be given.

Let x6 of type ι be given.

Let x7 of type ι be given.

Let x8 of type ι be given.

Let x9 of type ι be given.

Let x10 of type ι be given.

Let x11 of type ι be given.

Let x12 of type ι be given.

Let x13 of type ι be given.

Let x14 of type ι be given.

Let x15 of type ι be given.

Let x16 of type ι be given.

Apply H0 with x0, u17, λ x17 x18 . (λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x1 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x2 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x3 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x4 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x5 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x6 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x7 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x8 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x9 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x10 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x11 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x12 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x13 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x14 ((λ x19 x20 x21 x22 . If_i (x22 = x21) x19 x20) x15 x16 ((λ x19 : ι → ι . λ x20 . x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 x20)))))))))))))) ordsucc x17) x18) ((λ x19 : ι → ι . λ x20 . x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 x20))))))))))))) ordsucc x17) x18) ((λ x19 : ι → ι . λ x20 . x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 x20)))))))))))) ordsucc x17) x18) ((λ x19 : ι → ι . λ x20 . x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 (x19 x20))))))))))) ordsucc x17) x18) ... ...) ... ...) ... ...) ... ...) ... ...) ... ...) ... ...) ... ...) ... ...) ... ...) ... ..., 0.

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Proofgold Proof