const PNoLt : set (set prop) set (set prop) prop const PNoEq_ : set (set prop) (set prop) prop term PNoLe = \x:set.\p:set prop.\y:set.\q:set prop.PNoLt x p y q | x = y & PNoEq_ x p q const ordinal : set prop axiom PNoLtLe_tra: !x:set.!y:set.!z:set.ordinal x -> ordinal y -> ordinal z -> !p:set prop.!q:set prop.!p2:set prop.PNoLt x p y q -> PNoLe y q z p2 -> PNoLt x p z p2 lemma !x:set.!y:set.!z:set.!p:set prop.!q:set prop.!p2:set prop.ordinal y -> ordinal z -> PNoLe y q z p2 -> x = y -> PNoEq_ x p q -> PNoEq_ y p q -> PNoLe y p z p2 claim !x:set.!y:set.!z:set.ordinal x -> ordinal y -> ordinal z -> !p:set prop.!q:set prop.!p2:set prop.PNoLe x p y q -> PNoLe y q z p2 -> PNoLt x p z p2 | x = z & PNoEq_ x p p2