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 ordsucc : set set const omega : set axiom ordsucc_omega_ordinal: ordinal (ordsucc omega) const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom ordsuccI2: !x:set.x iIn ordsucc x axiom FalseE: ~ False axiom ordinal_In_Or_Subq: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | Subq y x const SNo_ : set set prop axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x const SNoS_ : set set axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x const SNoCut : set set set const binunion : set set set const famunion : set (set set) set const SNoEq_ : set set set prop axiom SNoCutP_SNoCut_impred: !x:set.!y:set.SNoCutP x y -> !P:prop.(SNo (SNoCut x y) -> SNoLev (SNoCut x y) iIn ordsucc (binunion (famunion x \z:set.ordsucc (SNoLev z)) (famunion y \z:set.ordsucc (SNoLev z))) -> (!z:set.z iIn x -> z < SNoCut x y) -> (!z:set.z iIn y -> SNoCut x y < z) -> (!z:set.SNo z -> (!w:set.w iIn x -> w < z) -> (!w:set.w iIn y -> z < w) -> Subq (SNoLev (SNoCut x y)) (SNoLev z) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) z) -> P) -> P const real : set const Empty : 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 const Sep : set (set prop) 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) -> SNo (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) -> SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) iIn ordsucc (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> y < SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) < y) -> ~ SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) iIn SNoS_ (ordsucc omega) lemma !x:set.(!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) -> SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) iIn ordsucc (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) -> Subq (ordsucc omega) (SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y))) -> ~ omega iIn SNoLev (SNoCut (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 hyp !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 claim ~ SNoCutP (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)