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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 binunion x0 (binintersect x1 x2), binintersect (binunion x0 x1) (binunion x0 x2) leaving 2 subgoals.
Apply binintersect_Subq_max with binunion x0 x1, binunion x0 x2, binunion x0 (binintersect x1 x2) leaving 2 subgoals.
Apply binunion_Subq_min with x0, binintersect x1 x2, binunion x0 x1 leaving 2 subgoals.
The subproof is completed by applying binunion_Subq_1 with x0, x1.
Apply Subq_tra with binintersect x1 x2, x1, binunion x0 x1 leaving 2 subgoals.
The subproof is completed by applying binintersect_Subq_1 with x1, x2.
The subproof is completed by applying binunion_Subq_2 with x0, x1.
Apply binunion_Subq_min with x0, binintersect x1 x2, binunion x0 x2 leaving 2 subgoals.
The subproof is completed by applying binunion_Subq_1 with x0, x2.
Apply Subq_tra with binintersect x1 x2, x2, binunion x0 x2 leaving 2 subgoals.
The subproof is completed by applying binintersect_Subq_2 with x1, x2.
The subproof is completed by applying binunion_Subq_2 with x0, x2.
Let x3 of type ι be given.
Apply binintersectE with binunion x0 x1, binunion x0 x2, x3, x3 ∈ binunion x0 (binintersect x1 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply binunionE with x0, x1, x3, x3 ∈ binunion x0 (binintersect x1 x2) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying binunionI1 with x0, binintersect x1 x2, x3. Assume H3: x3 ∈ x1.
Apply binunionE with x0, x2, x3, x3 ∈ binunion x0 (binintersect x1 x2) leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying binunionI1 with x0, binintersect x1 x2, x3. Assume H4: x3 ∈ x2.
Apply binunionI2 with x0, binintersect x1 x2, x3.
Apply binintersectI with x1, x2, x3 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
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