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 H2 with
x7,
x8,
λ x9 x10 . x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x10)))) = x1 x6 (x1 x2 (x1 x5 (x1 x8 (x1 x3 (x1 x4 x7))))) leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply unknownprop_b4da18a8864d7c18dd8366a9087a7bf26dfd188f3e8dc5d69726d033730ef2d6 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x8,
x7 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H9.
The subproof is completed by applying H8.