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Proofgold Proof

pf
Let x0 of type ι(ιιιο) → ιιο be given.
Assume H0: ∀ x1 . ∀ x2 x3 : ι → ι → ι → ο . (∀ x4 . x4x1x2 x4 = x3 x4)x0 x1 x2 = x0 x1 x3.
Apply In_ind with λ x1 . ∀ x2 x3 : ι → ι → ο . 6b023.. x0 x1 x26b023.. x0 x1 x3x2 = x3.
Let x1 of type ι be given.
Assume H1: ∀ x2 . x2x1∀ x3 x4 : ι → ι → ο . 6b023.. x0 x2 x36b023.. x0 x2 x4x3 = x4.
Let x2 of type ιιο be given.
Let x3 of type ιιο be given.
Assume H2: 6b023.. x0 x1 x2.
Assume H3: 6b023.. x0 x1 x3.
Claim L4: ∃ x4 : ι → ι → ι → ο . and (∀ x5 . x5x16b023.. x0 x5 (x4 x5)) (x2 = x0 x1 x4)
Apply unknownprop_d9a7c30bbc7df7f9c7418399a45d2dd0d50f90744e9b3fec6cc78c20d37defae with x0, x1, x2.
The subproof is completed by applying H2.
Claim L5: ∃ x4 : ι → ι → ι → ο . and (∀ x5 . x5x16b023.. x0 x5 (x4 x5)) (x3 = x0 x1 x4)
Apply unknownprop_d9a7c30bbc7df7f9c7418399a45d2dd0d50f90744e9b3fec6cc78c20d37defae with x0, x1, x3.
The subproof is completed by applying H3.
Apply exandE_iiio with λ x4 : ι → ι → ι → ο . ∀ x5 . x5x16b023.. x0 x5 (x4 x5), λ x4 : ι → ι → ι → ο . x2 = x0 x1 x4, x2 = x3 leaving 2 subgoals.
The subproof is completed by applying L4.
Let x4 of type ιιιο be given.
Assume H6: ∀ x5 . x5x16b023.. x0 x5 (x4 x5).
Assume H7: x2 = x0 x1 x4.
Apply exandE_iiio with λ x5 : ι → ι → ι → ο . ∀ x6 . x6x16b023.. x0 x6 (x5 x6), λ x5 : ι → ι → ι → ο . x3 = x0 x1 x5, x2 = x3 leaving 2 subgoals.
The subproof is completed by applying L5.
Let x5 of type ιιιο be given.
Assume H8: ∀ x6 . x6x16b023.. x0 x6 (x5 x6).
Assume H9: x3 = x0 x1 x5.
Apply H7 with λ x6 x7 : ι → ι → ο . x7 = x3.
Apply H9 with λ x6 x7 : ι → ι → ο . x0 x1 x4 = x7.
Apply H0 with x1, x4, x5.
Let x6 of type ι be given.
Assume H10: x6x1.
Apply H1 with x6, x4 x6, x5 x6 leaving 3 subgoals.
The subproof is completed by applying H10.
Apply H6 with x6.
The subproof is completed by applying H10.
Apply H8 with x6.
The subproof is completed by applying H10.