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 Empty : set axiom nat_0: nat_p 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) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const ordsucc : set set axiom nat_2: nat_p (ordsucc (ordsucc Empty)) axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const exp_SNo_nat : set set set axiom nat_exp_SNo_nat: !x:set.nat_p x -> !y:set.nat_p y -> nat_p (exp_SNo_nat x y) axiom SNoLt_irref: !x:set.~ x < x axiom FalseE: ~ False const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 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 abs_SNo_minus: !x:set.SNo x -> abs_SNo - x = abs_SNo x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_0L: !x:set.SNo x -> Empty + x = x axiom SNo_0: SNo Empty axiom SNoLt_0_1: Empty < ordsucc Empty const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom mul_SNo_zeroR: !x:set.SNo x -> x * Empty = Empty 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 SNo_omega: SNo omega const ordinal : set prop axiom omega_ordinal: ordinal omega axiom ordinal_In_SNoLt: !x:set.ordinal x -> !y:set.y iIn x -> y < x axiom minus_SNo_0: - Empty = Empty axiom minus_SNo_Lt_contra1: !x:set.!y:set.SNo x -> SNo y -> - x < y -> - y < x axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const real : set axiom SNoS_omega_real: Subq (SNoS_ omega) real 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 axiom real_I: !x:set.x iIn SNoS_ (ordsucc omega) -> x != omega -> x != - omega -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> x iIn real const SNoCut : set 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) -> SNo (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * 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) -> ~ SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) iIn real lemma !x:set.Empty < x -> SNo x -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> SNo (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) -> (!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) = omega -> Empty != x lemma !x:set.!y:set.Empty < x -> 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.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 omega -> eps_ y <= x -> nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) y) -> SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) < omega lemma !x:set.!y:set.x iIn real -> 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 -> y = SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) const binunion : set set set const famunion : set (set set) set 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 hyp SNo (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) hyp 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))) hyp !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) hyp !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 claim ~ SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) iIn SNoS_ (ordsucc omega)