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 (x1 x2 x3) x4 = x1 x2 (x1 x3 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
x2,
x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x8)))),
λ x9 x10 . x10 = x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 x2))))) leaving 3 subgoals.
The subproof is completed by applying H3.
Apply unknownprop_14619fcdadc5a43502995316176da02be54150d716fe5c9727e811d162c28b04 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8 leaving 7 subgoals.
The subproof is completed by applying H0.
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 H9.
Apply unknownprop_c4ddb10955189acaa0e73a79092d08be51c123dc1e586344a586ed01c4dfb456 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8,
x2 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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 H9.
The subproof is completed by applying H3.