Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Assume H0:
∀ x2 . In x2 x0 ⟶ ∀ x3 . In x3 x0 ⟶ In (x1 x2 x3) x0.
Let x2 of type ι → ι → ι → ι be given.
Assume H1:
∀ x3 . In x3 x0 ⟶ ∀ x4 . In x4 x0 ⟶ ∀ x5 . In x5 x0 ⟶ In (x2 x3 x4 x5) x0.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι → ι → ι be given.
Assume H4:
∀ x6 . In x6 x0 ⟶ ∀ x7 . In x7 x0 ⟶ In (x5 x6 x7) x0.
Let x6 of type ι → ι → ι be given.
Assume H5:
∀ x7 . In x7 x0 ⟶ ∀ x8 . In x8 x0 ⟶ In (x6 x7 x8) x0.
Let x7 of type ι be given.
Let x8 of type ι → ι → ι be given.
Assume H7:
∀ x9 . In x9 x0 ⟶ ∀ x10 . In x10 x0 ⟶ In (x8 x9 x10) x0.
Assume H8:
∀ x9 . In x9 x0 ⟶ (x8 x9 x7 = x9 ⟶ False) ⟶ False.
Assume H9:
∀ x9 . In x9 x0 ⟶ ∀ x10 . In x10 x0 ⟶ (x8 x9 (x1 x9 x10) = x10 ⟶ False) ⟶ False.
Assume H10:
∀ x9 . In x9 x0 ⟶ ∀ x10 . In x10 x0 ⟶ (x6 x9 x10 = x1 x9 (x8 x10 x9) ⟶ False) ⟶ False.
Assume H11:
∀ x9 . In x9 x0 ⟶ (x1 x7 x9 = x9 ⟶ False) ⟶ False.
Assume H12:
∀ x9 . In x9 x0 ⟶ (x5 x7 x9 = x9 ⟶ False) ⟶ False.
Assume H13:
∀ x9 . In x9 x0 ⟶ ∀ x10 . In x10 x0 ⟶ (x2 x9 x7 x10 = x10 ⟶ False) ⟶ False.
Assume H14:
∀ x9 . In x9 x0 ⟶ ∀ x10 . In x10 x0 ⟶ ∀ x11 . In x11 x0 ⟶ (x2 x9 x10 (x6 x9 (x5 x10 (x2 x9 x10 (x6 x9 (x5 x10 x11))))) = x11 ⟶ False) ⟶ False.
Assume H15:
∀ x9 . In x9 x0 ⟶ ∀ x10 . In x10 x0 ⟶ ∀ x11 . In x11 x0 ⟶ (x2 x9 x10 (x6 x9 (x6 x10 (x2 x9 x10 (x6 x9 (x6 x10 (x2 x9 x10 (x6 x9 (x6 x10 x11)))))))) = x11 ⟶ False) ⟶ False.
set y9 to be ...
set y10 to be ...
Assume H18:
x8 y9 y10 = x8 y10 y9 ⟶ False.
Apply H13 with
...,
... leaving 3 subgoals.