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Proofgold Proof

pf
Let x0 of type ιο be given.
Let x1 of type ιιι be given.
Assume H0: ∀ x2 x3 . x0 x2x0 x3x0 (x1 x2 x3).
Assume H1: ∀ x2 x3 x4 . x0 x2x0 x3x0 x4x1 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_2ce9a82c8ef9efc0240c60d5f07d019e2f7a44da8d6114bc529d6fb2d8f3a783 with x0, x1, x6, x3, x4, x5, x7, λ x8 x9 . x1 x2 x8 = x1 x6 (x1 x4 (x1 x3 (x1 x2 (x1 x5 x7)))) leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
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 H7.
Apply H1 with x2, x6, x1 x3 (x1 x4 (x1 x5 x7)), λ x8 x9 . x9 = x1 x6 (x1 x4 (x1 x3 (x1 x2 (x1 x5 x7)))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
Apply H0 with x3, x1 x4 (x1 x5 x7) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with x4, x1 x5 x7 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with x5, x7 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
set y8 to be x1 x6 (x1 x2 (x1 x3 (x1 x4 (x1 x5 x7))))
set y9 to be x2 x7 (x2 x5 (x2 x4 (x2 x3 (x2 x6 y8))))
Claim L8: ∀ x10 : ι → ο . x10 y9x10 y8
Let x10 of type ιο be given.
Assume H8: x10 (x3 y8 (x3 x6 (x3 x5 (x3 x4 (x3 x7 y9))))).
set y11 to be λ x11 . x10
Apply unknownprop_17f2e534568ee7312c417497530472991cbc191bc8362198ef82a32098ba0e8c with x2, x3, x4, x5, x6, x3 x7 y9, λ x12 x13 . y11 (x3 y8 x12) (x3 y8 x13) 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 H3.
The subproof is completed by applying H4.
Apply H0 with x7, y9 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Let x10 of type ιιο be given.
Apply L8 with λ x11 . x10 x11 y9x10 y9 x11.
Assume H9: x10 y9 y9.
The subproof is completed by applying H9.