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 ordsucc : set set const Empty : set axiom SNo_1: SNo (ordsucc Empty) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y axiom nonneg_mul_SNo_Le: !x:set.!y:set.!z:set.SNo x -> Empty <= x -> SNo y -> SNo z -> y <= z -> x * y <= x * z axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < 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 axiom FalseE: ~ False axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x const real : set const SNoS_ : set set const omega : 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 const Sep : set (set prop) set const SNoLev : 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) -> ~ SNoCutP (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) 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 hyp !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 claim ~ !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