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 const omega : set const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const Empty : set axiom SNoLt_minus_pos: !x:set.!y:set.SNo x -> SNo y -> x < y -> Empty < y + - x const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x axiom add_SNo_minus_Lt1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + - y) < z -> x < z + y axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom add_SNo_minus_Lt1b: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < z + y -> (x + - y) < z axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z axiom SNoLt_irref: !x:set.~ x < x const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const ordsucc : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoCut : set set set const Sep : set (set prop) set const SNoS_ : set set var x:set var y:set var z:set hyp x < ordsucc Empty hyp SNo x hyp SNo (SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) hyp !w:set.w iIn Sep (SNoS_ omega) (\u:set.ordsucc Empty < x * u) -> SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u) < w hyp y iIn SNoS_ omega hyp !w:set.w iIn omega -> abs_SNo (y + - SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u)) < eps_ w hyp SNo y hyp SNo (x * y) hyp SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < y hyp z iIn omega hyp ordsucc Empty <= x * y + - eps_ z claim ~ SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < y + - eps_ z