Let x0 of type ι be given.

Let x1 of type ι be given.

Apply set_ext with aae7a.. x0 x1, 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x2 . If_i (x2 = 4a7ef..) x0 x1) leaving 2 subgoals.

Let x2 of type ι be given.

Assume H0: prim1 x2 (aae7a.. x0 x1).

Apply unknownprop_583e189228469f510dae093aa816b0d084f1acaf0341e7deab9d9a676d1b11ef with x0, x1, x2, prim1 x2 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x3 . If_i (x3 = 4a7ef..) x0 x1)) leaving 3 subgoals.

The subproof is completed by applying H0.

Assume H1: ∃ x3 . and (prim1 x3 x0) (x2 = aae7a.. 4a7ef.. x3).

Apply exandE_i with λ x3 . prim1 x3 x0, λ x3 . x2 = aae7a.. 4a7ef.. x3, prim1 x2 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x3 . If_i (x3 = 4a7ef..) x0 x1)) leaving 2 subgoals.

The subproof is completed by applying H1.

Let x3 of type ι be given.

Assume H2: prim1 x3 x0.

Assume H3: x2 = aae7a.. 4a7ef.. x3.

Apply H3 with λ x4 x5 . prim1 x5 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x6 . If_i (x6 = 4a7ef..) x0 x1)).

Apply unknownprop_1f27075d0cd8d16888a609d68ca6246fb2307839dccadd646f85ab18bdcaae8e with 4ae4a.. (4ae4a.. 4a7ef..), λ x4 . If_i (x4 = 4a7ef..) x0 x1, 4a7ef.., x3 leaving 2 subgoals.

The subproof is completed by applying unknownprop_94c438c3f41134cd86e0be06a85b5e5b3aa8448f9221f51d2dfe9b1364042f49.

Apply If_i_1 with 4a7ef.. = 4a7ef.., x0, x1, λ x4 x5 . prim1 x3 x5 leaving 2 subgoals.

Let x4 of type ι → ι → ο be given.

Assume H4: x4 4a7ef.. 4a7ef...

The subproof is completed by applying H4.

The subproof is completed by applying H2.

Assume H1: ∃ x3 . and (prim1 x3 x1) (x2 = aae7a.. (4ae4a.. 4a7ef..) x3).

Apply exandE_i with λ x3 . prim1 x3 x1, λ x3 . x2 = aae7a.. (4ae4a.. 4a7ef..) x3, prim1 x2 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x3 . If_i (x3 = 4a7ef..) x0 x1)) leaving 2 subgoals.

The subproof is completed by applying H1.

Let x3 of type ι be given.

Assume H2: prim1 x3 x1.

Assume H3: x2 = aae7a.. (4ae4a.. 4a7ef..) x3.

Apply H3 with λ x4 x5 . prim1 x5 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x6 . If_i (x6 = 4a7ef..) x0 x1)).

Apply unknownprop_1f27075d0cd8d16888a609d68ca6246fb2307839dccadd646f85ab18bdcaae8e with 4ae4a.. (4ae4a.. 4a7ef..), λ x4 . If_i (x4 = 4a7ef..) x0 x1, 4ae4a.. 4a7ef.., x3 leaving 2 subgoals.

The subproof is completed by applying unknownprop_e256c3837ff221325e66d4c83283618d462d76cb96bca463e1abd4876bf63511.

Apply If_i_0 with 4ae4a.. 4a7ef.. = 4a7ef.., x0, x1, λ x4 x5 . prim1 x3 x5 leaving 2 subgoals.

The subproof is completed by applying unknownprop_24ff2ea632296eb0012bd83ffdc0e75761169422164b438efe0673b96d912be0.

The subproof is completed by applying H2.

Let x2 of type ι be given.

Assume H0: prim1 x2 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x3 . If_i (x3 = 4a7ef..) x0 x1)).

Claim L1: ...

...

Apply exandE_i with λ x3 . prim1 x3 (4ae4a.. (4ae4a.. 4a7ef..)), λ x3 . ∃ x4 . and (prim1 x4 (If_i (x3 = 4a7ef..) x0 x1)) (x2 = aae7a.. x3 x4), prim1 x2 (aae7a.. x0 x1) leaving 2 subgoals.

The subproof is completed by applying L1.

Let x3 of type ι be given.

Assume H2: prim1 x3 (4ae4a.. (4ae4a.. 4a7ef..)).

Assume H3: ∃ x4 . and (prim1 x4 (If_i (x3 = 4a7ef..) x0 x1)) (x2 = ...).

...

■

Proofgold Proof