Let x0 of type ι be given.
Let x1 of type (ι → ο) → ο be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι → ο be given.
Let x6 of type ι → ο be given.
Let x7 of type ι → ο be given.
Let x8 of type ι → ο be given.
Assume H0:
∀ x9 : ι → ο . (∀ x10 . x9 x10 ⟶ x10 ∈ x0) ⟶ iff (x1 x9) (x2 x9).
Assume H1: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x4 x9 x10.
Assume H2:
∀ x9 . x9 ∈ x0 ⟶ iff (x5 x9) (x6 x9).
Assume H3:
∀ x9 . x9 ∈ x0 ⟶ iff (x7 x9) (x8 x9).