const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w axiom SNoLt_irref: !x:set.~ x < x const SNoCut : set set set const Sep : set (set prop) set const SNoS_ : set set const omega : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const ordsucc : set set const Empty : set const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const eps_ : set set var x:set hyp Empty < x hyp SNo x hyp !y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x hyp SNo (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) hyp !y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) < y hyp SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) = omega claim Empty != x