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 H3:
PNoLt x0 x3 x1 x4.
Apply H4 with
PNoLt x0 x3 x2 x5 leaving 2 subgoals.
Assume H5:
PNoLt x1 x4 x2 x5.
Apply PNoLt_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 H3.
The subproof is completed by applying H5.
Apply H5 with
PNoLt x0 x3 x2 x5.
Assume H6: x1 = x2.
Apply H6 with
λ x6 x7 . PNoLt x0 x3 x6 x5.
Apply PNoLtEq_tra with
x0,
x1,
x3,
x4,
x5 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H7.