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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 (x1 x2 x3) 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.
Claim L7: ∀ x6 x7 x8 . x0 x6x0 x7x0 x8x1 x6 (x1 x7 x8) = x1 x7 (x1 x6 x8)
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Assume H7: x0 x6.
Assume H8: x0 x7.
Assume H9: x0 x8.
Apply H1 with x7, x6, x8, λ x9 x10 . x1 x6 (x1 x7 x8) = x10 leaving 4 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H7.
The subproof is completed by applying H9.
Apply H2 with x6, x7, λ x9 x10 . x1 x6 (x1 x7 x8) = x1 x9 x8 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H1 with x6, x7, x8 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply unknownprop_4febb148cd1e5d1398795072a8624a30b06608a0d196830eee8773b9d55c856a with x0, x1, x2, x3, x4, x5 leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L7.
The subproof is completed by applying H2.
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.