Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
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 H4:
PNoLe x1 x4 x2 x5.
Apply H3 with
PNoLe x0 x3 x2 x5 leaving 2 subgoals.
Assume H5:
PNoLt x0 x3 x1 x4.
Apply orIL with
PNoLt x0 x3 x2 x5,
and (x0 = x2) (PNoEq_ x0 x3 x5).
Apply PNoLtLe_tra with
x0,
x1,
x2,
x3,
x4,
x5 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
The subproof is completed by applying H4.
Apply H5 with
PNoLe x0 x3 x2 x5.
Assume H6: x0 = x1.
Apply H6 with
λ x6 x7 . PNoLe x7 x3 x2 x5.
Apply H6 with
λ x6 x7 . PNoEq_ x6 x3 x4.
The subproof is completed by applying H7.
Apply PNoEqLe_tra with
x1,
x2,
x3,
x4,
x5 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying L8.
The subproof is completed by applying H4.