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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Apply unknownprop_77b4d4df99acd26da01b3d17a9d3db16e9a142757eeed4d0180646664896d9b9 with
x0,
x1,
x5,
x2,
x3,
x4,
λ x6 x7 . x6 = x1 x5 (x1 x4 (x1 x3 x2)) leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply unknownprop_46d3e44975f3ecade481a2af0d8577c059537308401949cdec866cf7484652af with
x0,
x1,
x2,
x3,
x4,
λ x6 x7 . x1 x5 x7 = x1 x5 (x1 x4 (x1 x3 x2)) leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Let x6 of type ι → ι → ο be given.
Assume H7: x6 (x1 x5 (x1 x4 (x1 x3 x2))) (x1 x5 (x1 x4 (x1 x3 x2))).
The subproof is completed by applying H7.