Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
In x2 (lam x0 (λ x3 . x1 x3)).
Let x3 of type ι → ο be given.
Assume H1:
∀ x4 . In x4 x0 ⟶ ∀ x5 . In x5 (x1 x4) ⟶ x3 (lam 2 (λ x6 . If_i (x6 = 0) x4 x5)).
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x4 . In x4 x0,
λ x4 . ∃ x5 . and (In x5 (x1 x4)) (x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)),
x3 x2 leaving 2 subgoals.
Apply unknownprop_2defe3e203fb6cb62f6bbf233c5340c930ed4bb08b18129039b4272a28ee34c6 with
x0,
x1,
x2.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H3:
∃ x5 . and (In x5 (x1 x4)) (x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)).
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x5 . In x5 (x1 x4),
λ x5 . x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5),
x3 x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι be given.
Assume H4:
In x5 (x1 x4).
Assume H5:
x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5).
Apply H5 with
λ x6 x7 . x3 x7.
Apply H1 with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.