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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Apply unknownprop_515047c02fed97c50f69ea07f84c55a116d5435d48e441446058ba601add8797 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 . x9 = x1 x7 (x1 x6 (x1 x5 (x1 x4 (x1 x3 x2)))) 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 H8.
Apply H2 with
x2,
x7,
λ x8 x9 . x1 x3 (x1 x4 (x1 x5 (x1 x6 x9))) = x1 x7 (x1 x6 (x1 x5 (x1 x4 (x1 x3 x2)))) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H8.
Let x8 of type ι → ι → ο be given.
Apply unknownprop_c7b98fff279e959565508c1a89a93b6b521873fe12755031f6bf27d39fa3e6e1 with
x0,
x1,
x7,
x6,
x5,
x4,
x3,
x2,
λ x9 x10 . x8 x10 x9 leaving 8 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 H7.
The subproof is completed by applying H6.
The subproof is completed by applying H5.
The subproof is completed by applying H4.
The subproof is completed by applying H3.