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Proofgold Proof

pf
Let x0 of type ιο be given.
Let x1 of type ιιι be given.
Let x2 of type ιιι be given.
Assume H0: ∀ x3 x4 . x0 x3x0 x4x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 x5 . x0 x3x0 x4x0 x5x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
Assume H2: ∀ x3 x4 x5 . x0 x3x0 x4x0 x5x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5).
Assume H3: ∀ x3 x4 . x0 x3x0 x4x0 (x2 x3 x4).
Assume H4: ∀ x3 x4 x5 . x0 x3x0 x4x0 x5x1 (x1 x3 x4) x5 = x1 x3 (x1 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.
Assume H5: x0 x3.
Assume H6: x0 x4.
Assume H7: x0 x5.
Assume H8: x0 x6.
Apply unknownprop_72d123fd4d496da4fbf16cba456f7da0b762c033cc9f0bd9adf4e316000f1ad6 with x0, x1, x2, x3, x4, x5, x6, λ x7 x8 . x8 = x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x4 x5) (x2 x4 x6))) leaving 8 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 H6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H4 with x2 x3 x5, x2 x3 x6, x1 (x2 x4 x5) (x2 x4 x6) leaving 3 subgoals.
Apply H3 with x3, x5 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
Apply H3 with x3, x6 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
Apply H0 with x2 x4 x5, x2 x4 x6 leaving 2 subgoals.
Apply H3 with x4, x5 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
Apply H3 with x4, x6 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H8.