const Sep : set (set prop) set const SNoS_ : set set const ordsucc : set set const omega : set const minus_SNo : set set term - = minus_SNo const In : set set prop term iIn = In infix iIn 2000 2000 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set term real = Sep (SNoS_ (ordsucc omega)) \x:set.x != omega & x != - omega & !y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x const SNo : set prop axiom SNo_omega: SNo omega axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom FalseE: ~ False const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLeE: !x:set.!y:set.SNo x -> SNo y -> x <= y -> x < y | x = y const SNoLev : set set var x:set hyp x != - omega hyp SNoLev x iIn ordsucc omega hyp SNo x claim - omega <= x -> - omega < x