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Proofgold Proof

pf
Let x0 of type ι(ιιιο) → ιιο be given.
Let x1 of type ι be given.
Let x2 of type ιιο be given.
Assume H0: 6f52e.. x0 x1 x2.
Apply H0 with λ x3 . λ x4 : ι → ι → ο . ∃ x5 : ι → ι → ι → ο . and (∀ x6 . In x6 x36f52e.. x0 x6 (x5 x6)) (x4 = x0 x3 x5).
Let x3 of type ι be given.
Let x4 of type ιιιο be given.
Assume H1: ∀ x5 . In x5 x3∃ x6 : ι → ι → ι → ο . and (∀ x7 . In x7 x56f52e.. x0 x7 (x6 x7)) (x4 x5 = x0 x5 x6).
Let x5 of type ο be given.
Assume H2: ∀ x6 : ι → ι → ι → ο . and (∀ x7 . In x7 x36f52e.. x0 x7 (x6 x7)) (x0 x3 x4 = x0 x3 x6)x5.
Apply H2 with x4.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with ∀ x6 . In x6 x36f52e.. x0 x6 (x4 x6), x0 x3 x4 = x0 x3 x4 leaving 2 subgoals.
Let x6 of type ι be given.
Assume H3: In x6 x3.
Apply unknownprop_9ca1181159337344945687e86c7b1011fb442769c95e59cf3d126c226347254a with λ x7 : ι → ι → ι → ο . ∀ x8 . In x8 x66f52e.. x0 x8 (x7 x8), λ x7 : ι → ι → ι → ο . x4 x6 = x0 x6 x7, 6f52e.. x0 x6 (x4 x6) leaving 2 subgoals.
Apply H1 with x6.
The subproof is completed by applying H3.
Let x7 of type ιιιο be given.
Assume H4: ∀ x8 . In x8 x66f52e.. x0 x8 (x7 x8).
Assume H5: x4 x6 = x0 x6 x7.
Apply H5 with λ x8 x9 : ι → ι → ο . 6f52e.. x0 x6 x9.
Apply unknownprop_539172fd02c529de09e886b5940879e698f092d56b785ada1417814e30eb8f03 with x0, x6, x7.
The subproof is completed by applying H4.
Let x6 of type (ιιο) → (ιιο) → ο be given.
Assume H3: x6 (x0 x3 x4) (x0 x3 x4).
The subproof is completed by applying H3.