Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0:
∀ x3 . In x3 x0 ⟶ x1 x3 = x2 x3.
Apply unknownprop_c3fe42b21df0810041479a97b374de73f7754e07c8af1c88386a1e7dc0aad10f with
lam x0 (λ x3 . x1 x3),
lam x0 (λ x3 . x2 x3).
Let x3 of type ι be given.
Assume H1:
In x3 (lam x0 (λ x4 . x1 x4)).
Claim L2:
∃ x4 . and (In x4 x0) (∃ x5 . and (In x5 (x1 x4)) (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)))Apply unknownprop_2defe3e203fb6cb62f6bbf233c5340c930ed4bb08b18129039b4272a28ee34c6 with
x0,
x1,
x3.
The subproof is completed by applying H1.
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x4 . In x4 x0,
λ x4 . ∃ x5 . and (In x5 (x1 x4)) (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)),
In x3 (lam x0 (λ x4 . x2 x4)) leaving 2 subgoals.
The subproof is completed by applying L2.
Let x4 of type ι be given.
Assume H4:
∃ x5 . and (In x5 (x1 x4)) (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)).
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x5 . In x5 (x1 x4),
λ x5 . x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5),
In x3 (lam x0 (λ x5 . x2 x5)) leaving 2 subgoals.
The subproof is completed by applying H4.
Let x5 of type ι be given.
Assume H5:
In x5 (x1 x4).
Assume H6:
x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5).
Apply H6 with
λ x6 x7 . In x7 (lam x0 (λ x8 . x2 x8)).
Apply unknownprop_09484b774d66d459105c0920464e409b6d45859c91964dce5b8156bf6c4b7daa with
x0,
λ x6 . x2 x6,
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
λ x6 x7 . In x5 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.