Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Assume H0: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3).
Assume H1: ∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4).
Assume H2: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 x2 x3 = x1 x3 x2.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Assume H9: x0 x8.
Apply H1 with
x6,
x7,
x8,
λ x9 x10 . x1 x2 (x1 x3 (x1 x4 (x1 x5 x10))) = x1 x7 (x1 x3 (x1 x8 (x1 x5 (x1 x2 (x1 x4 x6))))) leaving 4 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply H2 with
x6,
x8,
λ x9 x10 . x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x7 x10)))) = x1 x7 (x1 x3 (x1 x8 (x1 x5 (x1 x2 (x1 x4 x6))))) leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H9.
Let x9 of type ι → ι → ο be given.
Apply unknownprop_ed23135aeb6396cf8942b08a749a97d26eba1158261651bcefec979027c92111 with
x0,
x1,
x7,
x3,
x8,
x5,
x2,
x4,
x6,
λ x10 x11 . x9 x11 x10 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H8.
The subproof is completed by applying H4.
The subproof is completed by applying H9.
The subproof is completed by applying H6.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H7.