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 → ι be given.
Assume H4:
∀ x6 . In x6 x0 ⟶ ∀ x7 . In x7 x0 ⟶ ∀ x8 . In x8 x0 ⟶ In (x5 x6 x7 x8) x0.
Let x6 of type ι be given.
Let x7 of type ι → ι → ι be given.
Assume H6:
∀ x8 . In x8 x0 ⟶ ∀ x9 . In x9 x0 ⟶ In (x7 x8 x9) x0.
Let x8 of type ι → ι → ι be given.
Assume H7:
∀ x9 . In x9 x0 ⟶ ∀ x10 . In x10 x0 ⟶ In (x8 x9 x10) x0.
Let x9 of type ι → ι → ι be given.
Assume H8:
∀ x10 . In x10 x0 ⟶ ∀ x11 . In x11 x0 ⟶ In (x9 x10 x11) x0.
Assume H9:
∀ x10 . In x10 x0 ⟶ ∀ x11 . In x11 x0 ⟶ (x9 x10 (x8 x10 x11) = x11 ⟶ False) ⟶ False.
Assume H10:
∀ x10 . In x10 x0 ⟶ ∀ x11 . In x11 x0 ⟶ (x1 x10 x11 = x8 x10 (x9 x11 x10) ⟶ False) ⟶ False.
Assume H11:
∀ x10 . In x10 x0 ⟶ (x7 x6 x10 = x10 ⟶ False) ⟶ False.
Assume H12:
∀ x10 . In x10 x0 ⟶ ∀ x11 . In x11 x0 ⟶ (x2 x6 x10 x11 = x11 ⟶ False) ⟶ False.
Assume H13:
∀ x10 . In x10 x0 ⟶ ∀ x11 . In x11 x0 ⟶ (x5 x6 x10 x11 = x11 ⟶ False) ⟶ False.
Assume H14:
∀ x10 . ... ⟶ ∀ x11 . In ... ... ⟶ ∀ x12 . In x12 x0 ⟶ ∀ x13 . In x13 x0 ⟶ (x5 x10 x12 (x1 x11 (x2 x10 x12 (x1 x11 (x5 x10 x12 (x1 x11 (x2 x10 x12 (x1 x11 (x5 x10 x12 (x1 x11 (x2 x10 x12 (x1 x11 (x5 x10 x12 (x1 x11 (x2 x10 x12 (x1 x11 (x5 x10 x12 (x1 x11 (x2 x10 x12 (x1 x11 x13))))))))))))))))))) = x13 ⟶ False) ⟶ False.