Let x0 of type ι → ο be given.
Assume H0:
∀ x1 . nat_p x1 ⟶ (∀ x2 . In x2 x1 ⟶ x0 x2) ⟶ x0 x1.
Claim L1:
∀ x1 . nat_p x1 ⟶ ∀ x2 . In x2 x1 ⟶ x0 x2Apply unknownprop_f23dde3020cfe827bdc4db0338b279dd2c0f6c90742a195a1a7a614475669076 with
λ x1 . ∀ x2 . In x2 x1 ⟶ x0 x2 leaving 2 subgoals.
Let x1 of type ι be given.
Apply FalseE with
x0 x1.
Apply unknownprop_1cc88f7e87aaf8c5cee24b4a69ff535a81e7855c45a9fd971eec05ee4cc28f9c with
x1.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Assume H2:
∀ x2 . In x2 x1 ⟶ x0 x2.
Let x2 of type ι be given.
Apply unknownprop_84fe37a922385756a4e0826a593defb788cadbe4bdc9a7fe6b519ea49f509df5 with
x1,
x2,
x0 x2 leaving 3 subgoals.
The subproof is completed by applying H3.
Apply H2 with
x2.
The subproof is completed by applying H4.
Assume H4: x2 = x1.
Apply H4 with
λ x3 x4 . x0 x4.
Apply H0 with
x1 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Apply H0 with
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply L1 with
x1.
The subproof is completed by applying H2.