Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι be given.
Let x4 of type ι → ι → ι be given.
Apply explicit_Nats_ind with
x0,
x1,
x2,
λ x5 . ∃ x6 . c3e2e.. x0 x1 x2 x3 x4 x5 x6 leaving 3 subgoals.
The subproof is completed by applying H0.
Let x5 of type ο be given.
Assume H1:
∀ x6 . c3e2e.. x0 x1 x2 x3 x4 x1 x6 ⟶ x5.
Apply H1 with
x3.
Let x6 of type ι → ι → ο be given.
Assume H2: x6 x1 x3.
Assume H3: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x6 x7 x8 ⟶ x6 (x2 x7) (x4 x7 x8).
The subproof is completed by applying H2.
Let x5 of type ι be given.
Assume H1: x5 ∈ x0.
Assume H2:
∃ x6 . c3e2e.. x0 x1 x2 x3 x4 x5 x6.
Apply H2 with
∃ x6 . c3e2e.. x0 x1 x2 x3 x4 (x2 x5) x6.
Let x6 of type ι be given.
Assume H3:
c3e2e.. x0 x1 x2 x3 x4 x5 x6.
Let x7 of type ο be given.
Assume H4:
∀ x8 . c3e2e.. x0 x1 x2 x3 x4 (x2 x5) x8 ⟶ x7.
Apply H4 with
x4 x5 x6.
Let x8 of type ι → ι → ο be given.
Assume H5: x8 x1 x3.
Assume H6: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x8 x9 x10 ⟶ x8 (x2 x9) (x4 x9 x10).
Apply H6 with
x5,
x6 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H3 with
x8 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.