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_6188398b9f7d79a13748e25a6502abdb9bec2af00da91608d0db3966f5053c1a with
x0,
x1,
x2,
ChurchNum2 x0 x3,
λ x4 x5 . x5 = ChurchNum2 x0 (ChurchNum2 x0 (x2 x3)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply unknownprop_cd8820c0941404bc0ee6e2d99bfa9cb130b650b8c5580df573b90c29b818d998 with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply unknownprop_6188398b9f7d79a13748e25a6502abdb9bec2af00da91608d0db3966f5053c1a with
x0,
x1,
x2,
x3,
λ x4 x5 . ChurchNum2 x0 x5 = ChurchNum2 x0 (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.