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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3.
Let x3 of type ι be given.
Assume H1: prim1 x3 (0fc90.. x0 (λ x4 . x1 x4)).
Claim L2: ∃ x4 . and (prim1 x4 x0) (∃ x5 . and (prim1 x5 (x1 x4)) (x3 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x6 . If_i (x6 = 4a7ef..) x4 x5)))
Apply unknownprop_b7e2d2f0cdd97ab9c73648950725dee9c8306169301c5c70b54b64bd81f587fb with x0, x1, x3.
The subproof is completed by applying H1.
Apply L2 with prim1 x3 (0fc90.. x0 x2).
Let x4 of type ι be given.
Assume H3: (λ x5 . and (prim1 x5 x0) (∃ x6 . and (prim1 x6 (x1 x5)) (x3 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x7 . If_i (x7 = 4a7ef..) x5 x6)))) x4.
Apply H3 with prim1 x3 (0fc90.. x0 x2).
Assume H4: prim1 x4 x0.
Assume H5: ∃ x5 . and (prim1 x5 (x1 x4)) (x3 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x6 . If_i (x6 = 4a7ef..) x4 x5)).
Apply H5 with prim1 x3 (0fc90.. x0 x2).
Let x5 of type ι be given.
Assume H6: (λ x6 . and (prim1 x6 (x1 x4)) (x3 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x7 . If_i (x7 = 4a7ef..) x4 x6))) x5.
Apply H6 with prim1 x3 (0fc90.. x0 x2).
Assume H7: prim1 x5 (x1 x4).
Assume H8: x3 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x6 . If_i (x6 = 4a7ef..) x4 x5).
Apply H8 with λ x6 x7 . prim1 x7 (0fc90.. x0 (λ x8 . x2 x8)).
Apply unknownprop_19a146fddf3209a9cb9037b3c55d31c340ac02a53a93d51ec1e8262af3504478 with x0, λ x6 . x2 x6, x4, x5 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with x4, λ x6 x7 . prim1 x5 x6 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
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