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_4aef431da355638d092d1af3952763e46a0de88399b3400cacc13c5390d4cf48 with
x0,
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x6 = x1 x3 (x1 x2 (x1 x4 x5)) leaving 7 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.
Apply unknownprop_4aef431da355638d092d1af3952763e46a0de88399b3400cacc13c5390d4cf48 with
x0,
x1,
x3,
x2,
x4,
x5,
λ x6 x7 . x1 (x1 x2 x3) (x1 x4 x5) = x6 leaving 7 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 H3.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
set y6 to be x1 (x1 x2 x3) (x1 x4 x5)
set y7 to be x2 (x2 x4 x3) (x2 x5 y6)
Claim L7: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6
Let x8 of type ι → ο be given.
Assume H7: x8 (x3 (x3 x5 x4) (x3 y6 y7)).
set y9 to be λ x9 . x8
Apply H2 with
x4,
x5,
λ x10 x11 . y9 (x3 x10 (x3 y6 y7)) (x3 x11 (x3 y6 y7)) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
Let x8 of type ι → ι → ο be given.
Apply L7 with
λ x9 . x8 x9 y7 ⟶ x8 y7 x9.
Assume H8: x8 y7 y7.
The subproof is completed by applying H8.