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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Apply unknownprop_ae303a8d0cab343f95e0be158bf945462f9c4cc9682540de91545edc76f520ff with
x0,
x1,
x4,
x2,
x3,
λ x5 x6 . 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 H5.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply H2 with
x2,
x3,
λ x5 x6 . x1 x4 x6 = x1 x4 (x1 x3 x2) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Let x5 of type ι → ι → ο be given.
Assume H6: x5 (x1 x4 (x1 x3 x2)) (x1 x4 (x1 x3 x2)).
The subproof is completed by applying H6.