Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Apply set_ext with 0ac37.. x0 (0ac37.. x1 x2), 0ac37.. (0ac37.. x0 x1) x2 leaving 2 subgoals.

Let x3 of type ι be given.

Assume H0: prim1 x3 (0ac37.. x0 (0ac37.. x1 x2)).

Apply unknownprop_b46721c187c37140cbae22d356b00ba89f4126d81d8665e4be15b5a58c78d06f with x0, 0ac37.. x1 x2, x3, prim1 x3 (0ac37.. (0ac37.. x0 x1) x2) leaving 3 subgoals.

The subproof is completed by applying H0.

Assume H1: prim1 x3 x0.

Apply unknownprop_0b5b61a66a1ed2eb843dbce5c5f6930c63a284fe5e33704d9f0cc564310ec40b with 0ac37.. x0 x1, x2, x3.

Apply unknownprop_0b5b61a66a1ed2eb843dbce5c5f6930c63a284fe5e33704d9f0cc564310ec40b with x0, x1, x3.

The subproof is completed by applying H1.

Assume H1: prim1 x3 (0ac37.. x1 x2).

Apply unknownprop_b46721c187c37140cbae22d356b00ba89f4126d81d8665e4be15b5a58c78d06f with x1, x2, x3, prim1 x3 (0ac37.. (0ac37.. x0 x1) x2) leaving 3 subgoals.

The subproof is completed by applying H1.

Assume H2: prim1 x3 x1.

Apply unknownprop_0b5b61a66a1ed2eb843dbce5c5f6930c63a284fe5e33704d9f0cc564310ec40b with 0ac37.. x0 x1, x2, x3.

Apply unknownprop_e4d6e0bfb4ef6d52ee13edd54a77c8cc7f0a3af8ffb1b8da66d4f98842dd28b5 with x0, x1, x3.

The subproof is completed by applying H2.

Assume H2: prim1 x3 x2.

Apply unknownprop_e4d6e0bfb4ef6d52ee13edd54a77c8cc7f0a3af8ffb1b8da66d4f98842dd28b5 with 0ac37.. x0 x1, x2, x3.

The subproof is completed by applying H2.

Let x3 of type ι be given.

Assume H0: prim1 x3 (0ac37.. (0ac37.. x0 x1) x2).

Apply unknownprop_b46721c187c37140cbae22d356b00ba89f4126d81d8665e4be15b5a58c78d06f with 0ac37.. x0 x1, x2, x3, prim1 x3 (0ac37.. x0 (0ac37.. x1 x2)) leaving 3 subgoals.

The subproof is completed by applying H0.

Assume H1: prim1 x3 (0ac37.. x0 x1).

Apply unknownprop_b46721c187c37140cbae22d356b00ba89f4126d81d8665e4be15b5a58c78d06f with x0, x1, x3, prim1 x3 (0ac37.. x0 (0ac37.. x1 x2)) leaving 3 subgoals.

The subproof is completed by applying H1.

Assume H2: prim1 x3 x0.

Apply unknownprop_0b5b61a66a1ed2eb843dbce5c5f6930c63a284fe5e33704d9f0cc564310ec40b with x0, 0ac37.. x1 x2, x3.

The subproof is completed by applying H2.

Assume H2: prim1 x3 x1.

Apply unknownprop_e4d6e0bfb4ef6d52ee13edd54a77c8cc7f0a3af8ffb1b8da66d4f98842dd28b5 with x0, 0ac37.. x1 x2, x3.

Apply unknownprop_0b5b61a66a1ed2eb843dbce5c5f6930c63a284fe5e33704d9f0cc564310ec40b with x1, x2, x3.

The subproof is completed by applying H2.

Assume H1: prim1 x3 x2.

Apply unknownprop_e4d6e0bfb4ef6d52ee13edd54a77c8cc7f0a3af8ffb1b8da66d4f98842dd28b5 with x0, 0ac37.. x1 x2, x3.

Apply unknownprop_e4d6e0bfb4ef6d52ee13edd54a77c8cc7f0a3af8ffb1b8da66d4f98842dd28b5 with x1, x2, x3.

The subproof is completed by applying H1.

■

Proofgold Proof