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.
Apply binintersectE with
x0,
binunion x1 x2,
x3,
x3 ∈ binunion (binintersect x0 x1) (binintersect x0 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3 ∈ x0.
Apply binunionE with
x1,
x2,
x3,
x3 ∈ binunion (binintersect x0 x1) (binintersect x0 x2) leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: x3 ∈ x1.
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: x3 ∈ x2.
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.