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 binintersect x0 (binunion x1 x2), binunion (binintersect x0 x1) (binintersect x0 x2) leaving 2 subgoals.

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

Let x3 of type ι be given.

Assume H0: In x3 (binintersect x0 (binunion x1 x2)).

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

The subproof is completed by applying H0.

Assume H1: In x3 x0.

Assume H2: In x3 (binunion x1 x2).

Apply unknownprop_a497a9c4fdb392b95b688b10c74f8f445a953a0c88030ccc02fa0b24e4758231 with x1, x2, x3, In x3 (binunion (binintersect x0 x1) (binintersect x0 x2)) leaving 3 subgoals.

The subproof is completed by applying H2.

Assume H3: In x3 x1.

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

Apply unknownprop_7e73699eda4c2a35af8db1aea1ddace7d2346405cd3944ace259823e1ec33cf3 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 binintersect 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_b493a75fcec57593a6576a7c6d850d3d24c9ed1931af4734bd2773dfce586a4d with x0, binunion x1 x2, binunion (binintersect x0 x1) (binintersect x0 x2) leaving 2 subgoals.

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

The subproof is completed by applying unknownprop_e0d452e0e0459ccd22d0a0ca66d8c6b520f69a835c0ebd50bbdd40077efd4a68 with x0, x1.

The subproof is completed by applying unknownprop_e0d452e0e0459ccd22d0a0ca66d8c6b520f69a835c0ebd50bbdd40077efd4a68 with x0, x2.

Apply unknownprop_fb26fe958e64ebbf533947db0048c8f1c2bfe1ee93c5358b327221e99f81f109 with binintersect x0 x1, binintersect x0 x2, binunion x1 x2 leaving 2 subgoals.

Apply unknownprop_16d2e0ac4e61f9d6f4710f68fd29bc0751fdb2a43ae4ee41dc39565d6502f8a7 with binintersect x0 x1, x1, binunion x1 x2 leaving 2 subgoals.

The subproof is completed by applying unknownprop_43a7b6a7250420244ab59ee561112cfc53c614b669837fc84c919b0df2ef6e0a with x0, x1.

The subproof is completed by applying unknownprop_45cad3d16e6a052331516a33ff7f695a27568e6c7992c9bc7ae67399a451422e with x1, x2.

Apply unknownprop_16d2e0ac4e61f9d6f4710f68fd29bc0751fdb2a43ae4ee41dc39565d6502f8a7 with binintersect x0 x2, x2, binunion x1 x2 leaving 2 subgoals.

The subproof is completed by applying unknownprop_43a7b6a7250420244ab59ee561112cfc53c614b669837fc84c919b0df2ef6e0a with x0, x2.

The subproof is completed by applying unknownprop_797168717b55ee5eded5eab45e11f1513074138c4f7a9fa70fbdac8b84ce4a03 with x1, x2.

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Proofgold Proof