Let x0 of type ι → ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x1 x3 ⟶ x1 (x0 x3).
Assume H1: ∀ x3 . x1 x3 ⟶ x2 (x0 x3) = x0 (x2 x3).
Let x3 of type ι be given.
Assume H2: x1 x3.
Apply unknownprop_1ecba3dd5eafa4f5c4681522469f6140788e091a78ad9f01e97cc0b6f2c0488f with
x0,
x1,
x2,
ChurchNum_ii_6 ChurchNum2 x0 x3,
λ x4 x5 . x5 = ChurchNum_ii_6 ChurchNum2 x0 (ChurchNum_ii_6 ChurchNum2 x0 (x2 x3)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply unknownprop_15156acfaef83289041d2fede437e15a29e8b8de66c5ea43e1171f433b1a0acd with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply unknownprop_1ecba3dd5eafa4f5c4681522469f6140788e091a78ad9f01e97cc0b6f2c0488f with
x0,
x1,
x2,
x3,
λ x4 x5 . ChurchNum_ii_6 ChurchNum2 x0 x5 = ChurchNum_ii_6 ChurchNum2 x0 (ChurchNum_ii_6 ChurchNum2 x0 (x2 x3)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ι → ι → ο be given.
The subproof is completed by applying H3.