Let x0 of type ι be given.

Assume H0: x0 ∈ complex.

Let x1 of type ο be given.

Assume H1: ∀ x2 . x2 ∈ real ⟶ ∀ x3 . x3 ∈ real ⟶ x0 = SNo_pair x2 x3 ⟶ x1.

Apply ReplE_impred with setprod real real, λ x2 . SNo_pair (ap x2 0) (ap x2 1), x0, x1 leaving 2 subgoals.

The subproof is completed by applying H0.

Let x2 of type ι be given.

Assume H2: x2 ∈ setprod real real.

Assume H3: x0 = SNo_pair (ap x2 0) (ap x2 1).

Apply H1 with ap x2 0, ap x2 1 leaving 3 subgoals.

Apply ap0_Sigma with real, λ x3 . real, x2.

The subproof is completed by applying H2.

Apply ap1_Sigma with real, λ x3 . real, x2.

The subproof is completed by applying H2.

The subproof is completed by applying H3.

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Proofgold Proof