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 omega : set const eps_ : set set const SNoS_ : set set axiom SNo_eps_SNoS_omega: !x:set.x iIn omega -> eps_ x iIn SNoS_ omega const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> !y:set.y iIn SNoS_ omega -> x + y iIn SNoS_ omega axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) 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) axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) axiom SNo_eps_pos: !x:set.x iIn omega -> Empty < eps_ x axiom mul_SNo_Lt1_pos_Lt: !x:set.!y:set.SNo x -> SNo y -> x < ordsucc Empty -> Empty < y -> x * y < y axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom mul_SNo_distrL: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * (y + z) = x * y + x * z const Sep : set (set prop) set axiom SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p 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 SNoLt_trichotomy_or_impred: !x:set.!y:set.SNo x -> SNo y -> !P:prop.(x < y -> P) -> (x = y -> P) -> (y < x -> P) -> P axiom FalseE: ~ False const minus_SNo : set set term - = minus_SNo axiom minus_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> - x iIn SNoS_ omega axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom minus_SNo_Lt_contra: !x:set.!y:set.SNo x -> SNo y -> x < y -> - y < - x axiom mul_SNo_minus_distrR: !x:set.!y:set.SNo x -> SNo y -> x * - y = - x * y axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z const SNoCut : set set set const abs_SNo : set set lemma !x:set.!y:set.!z:set.x < ordsucc Empty -> SNo x -> SNo (SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.x * u < ordsucc Empty) -> w < SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u)) -> y iIn SNoS_ omega -> (!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) -> SNo y -> SNo (x * y) -> y < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) -> z iIn omega -> (x * y + eps_ z) <= ordsucc Empty -> ~ (y + eps_ z) < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) lemma !x:set.!y:set.!z:set.x < ordsucc Empty -> SNo x -> SNo (SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) -> (!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) -> y iIn SNoS_ omega -> (!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) -> SNo y -> SNo (x * y) -> SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < y -> z iIn omega -> ordsucc Empty <= x * y + - eps_ z -> ~ SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < y + - eps_ z const real : set 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 hyp x * y < ordsucc Empty -> ?z:set.z iIn omega & (x * y + eps_ z) <= ordsucc Empty claim (ordsucc Empty < x * y -> ?z:set.z iIn omega & ordsucc Empty <= x * y + - eps_ z) -> y = SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)