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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ιι be given.
Let x4 of type ιι be given.
Assume H0: inj x0 x1 x3.
Assume H1: inj x1 x2 x4.
Apply H0 with inj x0 x2 (λ x5 . x4 (x3 x5)).
Assume H2: ∀ x5 . x5x0x3 x5x1.
Assume H3: ∀ x5 . x5x0∀ x6 . x6x0x3 x5 = x3 x6x5 = x6.
Apply H1 with inj x0 x2 (λ x5 . x4 (x3 x5)).
Assume H4: ∀ x5 . x5x1x4 x5x2.
Assume H5: ∀ x5 . x5x1∀ x6 . x6x1x4 x5 = x4 x6x5 = x6.
Apply andI with ∀ x5 . x5x0x4 (x3 x5)x2, ∀ x5 . x5x0∀ x6 . x6x0x4 (x3 x5) = x4 (x3 x6)x5 = x6 leaving 2 subgoals.
Let x5 of type ι be given.
Assume H6: x5x0.
Apply H4 with x3 x5.
Apply H2 with x5.
The subproof is completed by applying H6.
Let x5 of type ι be given.
Assume H6: x5x0.
Let x6 of type ι be given.
Assume H7: x6x0.
Assume H8: x4 (x3 x5) = x4 (x3 x6).
Apply H3 with x5, x6 leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
Apply H5 with x3 x5, x3 x6 leaving 3 subgoals.
Apply H2 with x5.
The subproof is completed by applying H6.
Apply H2 with x6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.