Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ο be given.
Apply unknownprop_4ad571f4a0c6aaf98e7cea1d7a5e094b177421e3e4501781303e29c4a79ec502 with
λ x3 x4 : ι → ο . x4 (Sep2 x0 x1 x2).
Let x3 of type ι be given.
Assume H0:
In x3 (Sep2 x0 x1 x2).
Apply unknownprop_806479d909a314585644562a92b76db0332e759a7bf7955909b140d76536eec6 with
x0,
x1,
x2,
x3,
λ x4 . ∃ x5 x6 . x4 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H2:
In x5 (x1 x4).
Assume H3: x2 x4 x5.
Let x6 of type ο be given.
Assume H4:
∀ x7 . (∃ x8 . lam 2 (λ x9 . If_i (x9 = 0) x4 x5) = lam 2 (λ x9 . If_i (x9 = 0) x7 x8)) ⟶ x6.
Apply H4 with
x4.
Let x7 of type ο be given.
Assume H5:
∀ x8 . lam 2 (λ x9 . If_i (x9 = 0) x4 x5) = lam 2 (λ x9 . If_i (x9 = 0) x4 x8) ⟶ x7.
Apply H5 with
x5.
Let x8 of type ι → ι → ο be given.
Assume H6:
x8 (lam 2 (λ x9 . If_i (x9 = 0) x4 x5)) (lam 2 (λ x9 . If_i (x9 = 0) x4 x5)).
The subproof is completed by applying H6.