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 nat_p : set prop const ordsucc : set set const Empty : set axiom nat_1: nat_p (ordsucc Empty) const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const SNoS_ : set set axiom omega_SNoS_omega: Subq omega (SNoS_ omega) axiom SNo_1: SNo (ordsucc Empty) const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo axiom SNoLt_minus_pos: !x:set.!y:set.SNo x -> SNo y -> x < y -> Empty < y + - x axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_minus_Le2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> z <= x + - y -> (z + y) <= x axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x axiom dneg: !P:prop.~ ~ P -> P const real : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoCut : set set set const Sep : set (set prop) set lemma !x:set.!y:set.x < ordsucc Empty -> ~ (?z:set.z iIn real & x * z = ordsucc Empty) -> SNo x -> SNo (SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)) -> (!z:set.z iIn Sep (SNoS_ omega) (\w:set.x * w < ordsucc Empty) -> z < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) -> (!z:set.z iIn Sep (SNoS_ omega) (\w:set.ordsucc Empty < x * w) -> SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < z) -> y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) < eps_ z) -> SNo y -> y iIn real -> SNo (x * y) -> (!z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x * y) < eps_ w) -> z = x * y) -> SNo - x * y -> (x * y < ordsucc Empty -> ?z:set.z iIn omega & (x * y + eps_ z) <= ordsucc Empty) -> y = SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) lemma !x:set.!y:set.SNo (x * y) -> (!z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x * y) < eps_ w) -> z = x * y) -> SNo - x * y -> x * y < ordsucc Empty -> ~ (?z:set.z iIn omega & (x * y + eps_ z) <= ordsucc Empty) -> ordsucc Empty != x * y var x:set var y:set hyp x < ordsucc Empty hyp ~ ?z:set.z iIn real & x * z = ordsucc Empty hyp SNo x hyp SNo (SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)) hyp !z:set.z iIn Sep (SNoS_ omega) (\w:set.x * w < ordsucc Empty) -> z < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) hyp !z:set.z iIn Sep (SNoS_ omega) (\w:set.ordsucc Empty < x * w) -> SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < z hyp y iIn SNoS_ omega hyp !z:set.z iIn omega -> abs_SNo (y + - SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) < eps_ z hyp SNo y hyp y iIn real hyp SNo (x * y) hyp !z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x * y) < eps_ w) -> z = x * y claim SNo - x * y -> y = SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)