const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const nat_p : set prop const ordinal : set prop axiom nat_p_ordinal: !x:set.nat_p x -> ordinal x axiom ordinal_ordsucc_In: !x:set.ordinal x -> !y:set.y iIn x -> ordsucc y iIn ordsucc x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const eps_ : set set axiom SNo_eps_decr: !x:set.x iIn omega -> !y:set.y iIn x -> eps_ x < eps_ y const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.nat_p x -> x iIn omega -> SNo (eps_ (ordsucc x)) -> y iIn x -> SNo (eps_ (ordsucc y)) -> eps_ (ordsucc x) < eps_ (ordsucc y) -> (eps_ (ordsucc x) + eps_ (ordsucc x)) < eps_ (ordsucc y) + eps_ (ordsucc y) var x:set var y:set hyp nat_p x hyp x iIn omega hyp SNo (eps_ (ordsucc x)) hyp y iIn x hyp y iIn omega claim SNo (eps_ (ordsucc y)) -> (eps_ (ordsucc x) + eps_ (ordsucc x)) < eps_ (ordsucc y) + eps_ (ordsucc y)