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.
Let x9 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.
Assume H10: x0 x9.
Apply H2 with
x8,
x9,
λ x10 x11 . x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x11))))) = x1 x5 (x1 x7 (x1 x2 (x1 x4 (x1 x3 (x1 x9 (x1 x6 x8)))))) leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply unknownprop_b0a7eeb7e0d195085dae477cfb5e17bfdce2df5680d52bbf6703cdd76bbee637 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x9,
x8 leaving 10 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 H8.
The subproof is completed by applying H10.
The subproof is completed by applying H9.