const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const equip : set set prop term finite = \x:set.?y:set.y iIn omega & equip x y term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const nat_p : set prop const ordinal : set prop axiom nat_p_ordinal: !x:set.nat_p x -> ordinal x const SNoS_ : set set lemma !x:set.x iIn omega -> nat_p x -> ordinal x -> finite (SNoS_ x) var x:set hyp x iIn omega claim nat_p x -> finite (SNoS_ x)