Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x2 x3 x4).
Assume H2: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x3 (x1 x4 x5) = x1 x4 (x1 x3 x5).
Assume H3: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 (x1 x3 x4) x5 = x1 x3 (x1 x4 x5).
Assume H4: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5).
Assume H5: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
Let x3 of type ι → ι be given.
Assume H6: ∀ x4 . x0 x4 ⟶ x3 x4 = x2 x4 x4.
Let x4 of type ι → ι be given.
Assume H7: ∀ x5 . x0 x5 ⟶ x0 (x4 x5).
Assume H8: ∀ x5 . x0 x5 ⟶ x4 (x4 x5) = x5.
Assume H9: ∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x1 (x4 x5) (x1 x5 x6) = x6.
Assume H10: ∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x1 x5 (x1 (x4 x5) x6) = x6.
Assume H11: ∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x2 (x4 x5) x6 = x4 (x2 x5 x6).
Assume H12: ∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x2 x5 (x4 x6) = x4 (x2 x5 x6).
Assume H13: ∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x2 x5 x6 = x2 x6 x5.
Assume H14: ∀ x5 x6 x7 x8 . x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x0 x8 ⟶ x2 (x2 x5 x6) (x2 x7 x8) = x2 (x2 x5 x7) (x2 x6 x8).
Let x5 of type ι → ο be given.
Assume H15:
∀ x6 . x0 x6 ⟶ ∃ x7 . and (x5 x7) (x3 x7 = x3 x6).
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H16: x0 x6.
Assume H17: x0 x7.
Let x8 of type ι be given.
Let x9 of type ι be given.
Let x10 of type ι be given.
Let x11 of type ι be given.
Assume H18: x0 x8.
Assume H19: x0 x9.
Assume H20: x0 x10.
Assume H21: x0 x11.
Assume H22: x6 = x1 (x3 x8) (x1 (x3 x9) (x1 ... ...)).