Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: prim1 x1 x2.

Assume H1: prim1 x2 x0.

Apply UnionEq with x0, x1, prim1 x1 (prim3 x0).

Assume H2: prim1 x1 (prim3 x0) ⟶ ∃ x3 . and (prim1 x1 x3) (prim1 x3 x0).

Assume H3: (∃ x3 . and (prim1 x1 x3) (prim1 x3 x0)) ⟶ prim1 x1 (prim3 x0).

Apply H3.

Let x3 of type ο be given.

Assume H4: ∀ x4 . and (prim1 x1 x4) (prim1 x4 x0) ⟶ x3.

Apply H4 with x2.

Apply andI with prim1 x1 x2, prim1 x2 x0 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

■

Proofgold Proof