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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 ιιι be given.
Let x4 of type ι be given.
Let x5 of type ιιι be given.
Let x6 of type ιιιι be given.
Let x7 of type ιιι be given.
Let x8 of type ιιιι be given.
Let x9 of type ιιιι be given.
Let x10 of type ιιι be given.
Let x11 of type ιιι be given.
Let x12 of type ιιι be given.
Let x13 of type ιιι be given.
Assume H0: Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
Assume H1: In x4 x0.
Assume H2: ∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 x16)))))))))))))) = x16.
Assume H3: ∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 x16)))))))) = x16.
Assume H4: ∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x7 x15 (x12 x16 x17)) = x7 x15 (x12 x16 (x12 x14 x17)).
Assume H5: ∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x13 x17 x18)) = x12 x16 (x13 x17 (x9 x14 x15 x18)).
Assume H6: ∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x12 x14 (x12 x15 x18))).
Assume H7: ∀ x14 . ...∀ x15 . ...∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x12 x14 (x10 x15 x18))).
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