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 ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const SNoS_ : set set const SNoLev : set set const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P const Sep : set (set prop) set axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y const real : set const Empty : set const ordsucc : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 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 lemma !x:set.x iIn real -> Empty < x -> x < ordsucc Empty -> ~ (?y:set.y iIn real & x * y = ordsucc Empty) -> SNo x -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> !P:prop.(SNo y -> SNoLev y iIn omega -> x * y < ordsucc Empty -> P) -> P) -> ~ !y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> !P:prop.(SNo y -> SNoLev y iIn omega -> ordsucc Empty < x * y -> P) -> P var x:set hyp x iIn real hyp Empty < x hyp x < ordsucc Empty hyp ~ ?y:set.y iIn real & x * y = ordsucc Empty hyp SNo x hyp !y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x claim ~ !y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> !P:prop.(SNo y -> SNoLev y iIn omega -> x * y < ordsucc Empty -> P) -> P