Let x0 of type ι be given.
Let x1 of type ι be given.
Apply prop_ext_2 with
x0 ⊆ x1,
binintersect x0 x1 = x0 leaving 2 subgoals.
Assume H0: x0 ⊆ x1.
Apply set_ext with
binintersect x0 x1,
x0 leaving 2 subgoals.
The subproof is completed by applying binintersect_Subq_1 with x0, x1.
Apply binintersect_Subq_max with
x0,
x1,
x0 leaving 2 subgoals.
The subproof is completed by applying Subq_ref with x0.
The subproof is completed by applying H0.
Apply H0 with
λ x2 x3 . x2 ⊆ x1.
The subproof is completed by applying binintersect_Subq_2 with x0, x1.