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).
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 H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Apply unknownprop_9551c74ff689713f6e29d6039e164b7f427808e036d50b6ddb9edf722f975820 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 . x1 x2 x9 = x1 x5 (x1 x2 (x1 x6 (x1 x3 (x1 x4 x7)))) leaving 8 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.
Apply H1 with
x2,
x5,
x1 x6 (x1 x3 (x1 x4 x7)),
λ x8 x9 . x9 = x1 x5 (x1 x2 (x1 x6 (x1 x3 (x1 x4 x7)))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x3 (x1 x4 x7) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x3,
x1 x4 x7 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x7 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
Let x8 of type ι → ι → ο be given.
Assume H8: x8 (x1 x5 (x1 x2 (x1 x6 (x1 x3 (x1 x4 x7))))) (x1 x5 (x1 x2 (x1 x6 (x1 x3 (x1 x4 x7))))).
The subproof is completed by applying H8.