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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0: ordinal x0.
Assume H1: ordinal x1.
Assume H2: ordinal x2.
Let x3 of type ιο be given.
Let x4 of type ιο be given.
Let x5 of type ιο be given.
Assume H3: PNoLt x0 x3 x1 x4.
Assume H4: or (PNoLt x1 x4 x2 x5) (and (x1 = x2) (PNoEq_ x1 x4 x5)).
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.
Assume H5: and (x1 = x2) (PNoEq_ x1 x4 x5).
Apply H5 with PNoLt x0 x3 x2 x5.
Assume H6: x1 = x2.
Assume H7: PNoEq_ x1 x4 x5.
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.