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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιιι be given.
Let x2 of type ιιι be given.
Let x3 of type ι be given.
Assume H0: 06179.. (e707a.. x0 x1 x2 x3).
Apply H0 with λ x4 . x4 = e707a.. x0 x1 x2 x3∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0prim1 (x1 x5 x6) x0 leaving 2 subgoals.
Let x4 of type ι be given.
Let x5 of type ιιι be given.
Assume H1: ∀ x6 . prim1 x6 x4∀ x7 . prim1 x7 x4prim1 (x5 x6 x7) x4.
Let x6 of type ιιι be given.
Assume H2: ∀ x7 . prim1 x7 x4∀ x8 . prim1 x8 x4prim1 (x6 x7 x8) x4.
Let x7 of type ι be given.
Assume H3: prim1 x7 x4.
Assume H4: e707a.. x4 x5 x6 x7 = e707a.. x0 x1 x2 x3.
Apply unknownprop_dd5c8ac7766b5ba866e12a521f09590f56b29aeb96649355d90f224c93aa6f7c with x4, x0, x5, x1, x6, x2, x7, x3, ∀ x8 . prim1 x8 x0∀ x9 . prim1 x9 x0prim1 (x1 x8 x9) x0 leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H5: and (and (x4 = x0) (∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x5 x8 x9 = x1 x8 x9)) (∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x6 x8 x9 = x2 x8 x9).
Apply H5 with x7 = x3∀ x8 . prim1 x8 x0∀ x9 . prim1 x9 x0prim1 (x1 x8 x9) x0.
Assume H6: and (x4 = x0) (∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x5 x8 x9 = x1 x8 x9).
Apply H6 with (∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x6 x8 x9 = x2 x8 x9)x7 = x3∀ x8 . prim1 x8 x0∀ x9 . prim1 x9 x0prim1 (x1 x8 x9) x0.
Assume H7: x4 = x0.
Assume H8: ∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x5 x8 x9 = x1 x8 x9.
Assume H9: ∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x6 x8 x9 = x2 x8 x9.
Assume H10: x7 = x3.
Apply H7 with λ x8 x9 . ∀ x10 . prim1 x10 x8∀ x11 . prim1 x11 x8prim1 (x1 x10 x11) x8.
Let x8 of type ι be given.
Assume H11: prim1 x8 x4.
Let x9 of type ι be given.
Assume H12: prim1 x9 x4.
Apply H8 with x8, x9, λ x10 x11 . prim1 x10 ... leaving 3 subgoals.
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