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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: ordinal x0.
Let x1 of type ιο be given.
Let x2 of type ι be given.
Assume H1: prim1 x2 x0.
Apply iffI with prim1 x2 (09072.. x0 x1), x1 x2 leaving 2 subgoals.
Assume H2: prim1 x2 (0ac37.. (1216a.. x0 (λ x3 . x1 x3)) (a4c2a.. x0 (λ x3 . not (x1 x3)) (λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3))).
Apply unknownprop_b46721c187c37140cbae22d356b00ba89f4126d81d8665e4be15b5a58c78d06f with 1216a.. x0 (λ x3 . x1 x3), a4c2a.. x0 (λ x3 . not (x1 x3)) (λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3), x2, x1 x2 leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: prim1 x2 (1216a.. x0 (λ x3 . x1 x3)).
Apply unknownprop_e75b5686f39ea4dca8e72e616b0514162494e9e895f52dbe14fa1984a713fe57 with x0, x1, x2.
The subproof is completed by applying H3.
Assume H3: prim1 x2 (a4c2a.. x0 (λ x3 . not (x1 x3)) (λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3)).
Apply FalseE with x1 x2.
Apply unknownprop_e546e9a8cc28c7314a8604ada98e2a83641f2ef6b8078441570ffe037b28d26f with x0, λ x3 . not (x1 x3), λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3, x2, False leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: prim1 x3 x0.
Assume H5: not (x1 x3).
Assume H6: x2 = (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3.
Apply unknownprop_d50cf015ad4bd21633c5a2d035ded36b98ffbab6f00fb800e392ce4bcc7ee247 with x0, x3 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply H6 with λ x4 x5 . prim1 x4 x0.
The subproof is completed by applying H1.
Assume H2: x1 x2.
Apply unknownprop_0b5b61a66a1ed2eb843dbce5c5f6930c63a284fe5e33704d9f0cc564310ec40b with 1216a.. x0 (λ x3 . x1 x3), a4c2a.. x0 (λ x3 . not (x1 x3)) (λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3), x2.
Apply unknownprop_1dada0fb38ff7f9b45b564ad11d6295d93250427446875218f17ee62431454a6 with x0, λ x3 . x1 x3, x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.