Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Apply unknownprop_a23ec6a55ac212526d74cbf0d04096929ad453b0eb0f8023e32b8a33930d39fb with setminus x0 (setminus x1 x2), binunion (setminus x0 x1) (binintersect x0 x2) leaving 2 subgoals.

Apply unknownprop_c3fe42b21df0810041479a97b374de73f7754e07c8af1c88386a1e7dc0aad10f with setminus x0 (setminus x1 x2), binunion (setminus x0 x1) (binintersect x0 x2).

Let x3 of type ι be given.

Assume H0: In x3 (setminus x0 (setminus x1 x2)).

Apply unknownprop_e02b92c94ff70655b8eb0623a7ec106c0c0c9c65ac0f52f9689f2e6c9f563b5d with x0, setminus x1 x2, x3, In x3 (binunion (setminus x0 x1) (binintersect x0 x2)) leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H1: In x3 x0.

Assume H2: nIn x3 (setminus x1 x2).

Apply unknownprop_1afd64ac8b0801db96bac383a6886e922ca3e0f7a15b76edf96b4d4ee849f6d4 with x1, x2, x3, In x3 (binunion (setminus x0 x1) (binintersect x0 x2)) leaving 3 subgoals.

The subproof is completed by applying H2.

Assume H3: nIn x3 x1.

Apply unknownprop_1598d20a62a395ced156dfcc7d767ab023594ea6ef7c5e3b53cecdbebaf0ec29 with setminus x0 x1, binintersect x0 x2, x3.

Apply unknownprop_3fe7f97dcb00bcf31cf989081bd8403f8f0647acc7dd719f8a7da64cd4837dac with x0, x1, x3 leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H3.

Assume H3: In x3 x2.

Apply unknownprop_e6919c45bef19e01f88ce072c705412578331e9a3c7532de752ffb4187ed1265 with setminus x0 x1, binintersect x0 x2, x3.

Apply unknownprop_7e73699eda4c2a35af8db1aea1ddace7d2346405cd3944ace259823e1ec33cf3 with x0, x2, x3 leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H3.

Apply unknownprop_fb26fe958e64ebbf533947db0048c8f1c2bfe1ee93c5358b327221e99f81f109 with setminus x0 x1, binintersect x0 x2, setminus x0 (setminus x1 x2) leaving 2 subgoals.

Apply unknownprop_c3fe42b21df0810041479a97b374de73f7754e07c8af1c88386a1e7dc0aad10f with setminus x0 x1, setminus x0 (setminus x1 x2).

Let x3 of type ι be given.

Assume H0: In x3 (setminus x0 x1).

Apply unknownprop_e02b92c94ff70655b8eb0623a7ec106c0c0c9c65ac0f52f9689f2e6c9f563b5d with x0, x1, x3, In x3 (setminus x0 (setminus x1 x2)) leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H1: In x3 x0.

Assume H2: nIn x3 x1.

Apply unknownprop_3fe7f97dcb00bcf31cf989081bd8403f8f0647acc7dd719f8a7da64cd4837dac with x0, setminus x1 x2, x3 leaving 2 subgoals.

The subproof is completed by applying H1.

Apply unknownprop_478209dd41b4519febe499dfc419570987408946f95d28730c404b54b09f0a65 with x1, x2, x3.

The subproof is completed by applying H2.

Apply unknownprop_c3fe42b21df0810041479a97b374de73f7754e07c8af1c88386a1e7dc0aad10f with binintersect x0 x2, setminus x0 (setminus x1 x2).

Let x3 of type ι be given.

Assume H0: In x3 (binintersect x0 x2).

Apply unknownprop_9f9c1680d203bdbb862d1bf6c2b8504d7e3a6fca72f77bd8968e86ad6ad69346 with x0, x2, x3, In x3 (setminus x0 (setminus x1 x2)) leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H1: In x3 x0.

Assume H2: In x3 x2.

Apply unknownprop_3fe7f97dcb00bcf31cf989081bd8403f8f0647acc7dd719f8a7da64cd4837dac with x0, setminus x1 x2, x3 leaving 2 subgoals.

The subproof is completed by applying H1.

Apply unknownprop_73a703328ee04d26917cf0d36cf8873ca8eee5e7d06481a703a2fc5b325368e7 with x1, x2, x3.

The subproof is completed by applying H2.

■

Proofgold Proof