Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply H0 with
∀ x3 x4 x5 : ι → ο . PNoLt x0 x3 x1 x4 ⟶ PNoLt x1 x4 x2 x5 ⟶ PNoLt x0 x3 x2 x5.
Assume H4:
∀ x3 . x3 ∈ x0 ⟶ TransSet x3.
Apply H2 with
∀ x3 x4 x5 : ι → ο . PNoLt x0 x3 x1 x4 ⟶ PNoLt x1 x4 x2 x5 ⟶ PNoLt x0 x3 x2 x5.
Assume H6:
∀ x3 . x3 ∈ x2 ⟶ TransSet x3.
Let x3 of type ι → ο be given.
Let x4 of type ι → ο be given.
Let x5 of type ι → ο be given.
Assume H7:
PNoLt x0 x3 x1 x4.
Assume H8:
PNoLt x1 x4 x2 x5.
Apply PNoLtE with
x0,
x1,
x3,
x4,
PNoLt x0 x3 x2 x5 leaving 4 subgoals.
The subproof is completed by applying H7.
Apply H9 with
PNoLt x0 x3 x2 x5.
Let x6 of type ι be given.
Apply H10 with
PNoLt x0 x3 x2 x5.
Apply binintersectE with
x0,
x1,
x6,
and (and (PNoEq_ x6 x3 x4) (not (x3 x6))) (x4 x6) ⟶ PNoLt x0 x3 x2 x5 leaving 2 subgoals.
The subproof is completed by applying H11.
Assume H12: x6 ∈ x0.
Assume H13: x6 ∈ x1.
Apply H14 with
PNoLt x0 x3 x2 x5.
Apply H15 with
x4 x6 ⟶ PNoLt x0 x3 x2 x5.
Assume H18: x4 x6.
Apply PNoLtE with
x1,
x2,
x4,
x5,
PNoLt x0 x3 x2 x5 leaving 4 subgoals.
The subproof is completed by applying H8.
Apply H20 with
PNoLt x0 x3 x2 x5.
Let x7 of type ι be given.
Apply H21 with
PNoLt x0 x3 x2 x5.
Apply binintersectE with
x1,
x2,
x7,
and (and (PNoEq_ x7 x4 x5) (not (x4 x7))) ... ⟶ PNoLt x0 x3 x2 x5 leaving 2 subgoals.