Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ο be given.
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 . and (In x5 x0) (∃ x6 . and (In x6 (x1 x5)) (and (x4 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 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 . and (In x7 x0) (∃ x8 . and (In x8 (x1 x7)) (and (lam 2 (λ x9 . If_i (x9 = 0) x4 x5) = lam 2 (λ x9 . If_i (x9 = 0) x7 x8)) (x2 x7 x8))) ⟶ x6.
Apply H4 with
x4.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x4 x0,
∃ x7 . and (In x7 (x1 x4)) (and (lam 2 (λ x8 . If_i (x8 = 0) x4 x5) = lam 2 (λ x8 . If_i (x8 = 0) x4 x7)) (x2 x4 x7)) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x7 of type ο be given.
Assume H5:
∀ x8 . and (In x8 (x1 x4)) (and (lam 2 (λ x9 . If_i (x9 = 0) x4 x5) = lam 2 (λ x9 . If_i (x9 = 0) x4 x8)) (x2 x4 x8)) ⟶ x7.
Apply H5 with
x5.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x5 (x1 x4),
and (lam 2 (λ x8 . If_i (x8 = 0) x4 x5) = lam 2 (λ x8 . If_i (x8 = 0) x4 x5)) (x2 x4 x5) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
lam 2 (λ x8 . If_i (x8 = 0) x4 x5) = lam 2 (λ x8 . If_i (x8 = 0) x4 x5),
x2 x4 x5 leaving 2 subgoals.
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.
The subproof is completed by applying H3.