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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).
Assume H2: ∀ x2 x3 . x0 x2x0 x3x1 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 H2 with x4, x5, λ x6 x7 . x1 x2 (x1 x3 x7) = x1 x5 (x1 x2 (x1 x3 x4)) leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Let x6 of type ιιο be given.
Apply unknownprop_6df806693864a23a378ddbca02cda4bb4bc233ff1daa8914d51c06eb72ff2550 with x0, x1, x5, x2, x3, x4, λ x7 x8 . x6 x8 x7 leaving 6 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.