Let x0 of type ι be given.

Let x1 of type ι → ι → ο be given.

Assume H0: ∀ x2 . In x2 x0 ⟶ ∃ x3 . x1 x2 x3.

Let x2 of type ο be given.

Assume H1: ∀ x3 : ι → ι . (∀ x4 . In x4 x0 ⟶ x1 x4 (x3 x4)) ⟶ x2.

Apply H1 with λ x3 . Eps_i (x1 x3).

Let x3 of type ι be given.

Assume H2: In x3 x0.

Apply H0 with x3, x1 x3 (Eps_i (x1 x3)) leaving 2 subgoals.

The subproof is completed by applying H2.

Let x4 of type ι be given.

Assume H3: x1 x3 x4.

Apply unknownprop_c3f0de4cb966012957ca752938aa96a32c594389e7aea45227d571c0506618ba with x1 x3, x4.

The subproof is completed by applying H3.

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Proofgold Proof