Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
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 x3.
Assume H3: x0 x4.
Assume H4: x0 x5.
Assume H5: x0 x6.
Assume H6: x0 x7.
Apply H1 with
x3,
x1 x4 (x1 x5 x6),
x7,
λ x8 x9 . x9 = x1 (x2 x3 x7) (x1 (x2 x4 x7) (x1 (x2 x5 x7) (x2 x6 x7))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_a6006624829d7d44dfef146f097f47f429351018787d2b11ce9751df54eb1332 with
x0,
x1,
x4,
x5,
x6 leaving 4 subgoals.
The subproof is completed by applying H0.
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.
set y8 to be x1 (x2 x3 x7) (x2 (x1 x4 (x1 x5 x6)) x7)
set y9 to be x2 (x3 x4 y8) (x2 (x3 x5 y8) (x2 (x3 x6 y8) (x3 x7 y8)))
Claim L7: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H7: x10 (x3 (x4 x5 y9) (x3 (x4 x6 y9) (x3 (x4 x7 y9) (x4 y8 y9)))).
set y11 to be λ x11 . x10
Apply unknownprop_39e817a8f257892486a787991782a9298ace278e00bb99d6258d016dbbcaeb22 with
x2,
x3,
x4,
x6,
x7,
y8,
y9,
λ x12 x13 . y11 (x3 (x4 x5 y9) x12) (x3 (x4 x5 y9) x13) 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.
The subproof is completed by applying H7.
Let x10 of type ι → ι → ο be given.
Apply L7 with
λ x11 . x10 x11 y9 ⟶ x10 y9 x11.
Assume H8: x10 y9 y9.
The subproof is completed by applying H8.