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Proofgold Proof

pf
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 binintersect x0 (binunion x1 x2), binunion (binintersect x0 x1) (binintersect x0 x2) leaving 2 subgoals.
Let x3 of type ι be given.
Assume H0: x3binintersect x0 (binunion x1 x2).
Apply binintersectE with x0, binunion x1 x2, x3, x3binunion (binintersect x0 x1) (binintersect x0 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3x0.
Assume H2: x3binunion x1 x2.
Apply binunionE with x1, x2, x3, x3binunion (binintersect x0 x1) (binintersect x0 x2) leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: x3x1.
Apply binunionI1 with binintersect x0 x1, binintersect x0 x2, x3.
Apply binintersectI with x0, x1, x3 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Assume H3: x3x2.
Apply binunionI2 with binintersect x0 x1, binintersect x0 x2, x3.
Apply binintersectI with x0, x2, x3 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply binintersect_Subq_max with x0, binunion x1 x2, binunion (binintersect x0 x1) (binintersect x0 x2) leaving 2 subgoals.
Apply binunion_Subq_min with binintersect x0 x1, binintersect x0 x2, x0 leaving 2 subgoals.
The subproof is completed by applying binintersect_Subq_1 with x0, x1.
The subproof is completed by applying binintersect_Subq_1 with x0, x2.
Apply binunion_Subq_min with binintersect x0 x1, binintersect x0 x2, binunion x1 x2 leaving 2 subgoals.
Apply Subq_tra with binintersect x0 x1, x1, binunion x1 x2 leaving 2 subgoals.
The subproof is completed by applying binintersect_Subq_2 with x0, x1.
The subproof is completed by applying binunion_Subq_1 with x1, x2.
Apply Subq_tra with binintersect x0 x2, x2, binunion x1 x2 leaving 2 subgoals.
The subproof is completed by applying binintersect_Subq_2 with x0, x2.
The subproof is completed by applying binunion_Subq_2 with x1, x2.