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 term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x 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 term nIn = \x:set.\y:set.~ x iIn y const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const omega : set 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 minus_SNo_Le_contra: !x:set.!y:set.SNo x -> SNo y -> x <= y -> - y <= - x axiom add_SNo_Le2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y <= z -> (x + y) <= x + z const Empty : set const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x 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 Sep : set (set prop) set const SNoS_ : set set const ap : set set set var x:set var y:set var z:set var w:set var u:set hyp !v:set.v iIn omega -> SNo (ap y v) hyp SNo (SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v)) hyp !v:set.v iIn omega -> abs_SNo (z + - SNoCut (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & x2 < ap x y2) (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & ap y y2 < x2)) < eps_ v hyp SNo z hyp w iIn omega hyp ap y w < z hyp u iIn omega hyp Empty < z + - ap y w hyp SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v) <= ap y w claim Empty < z + - SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v) -> abs_SNo (z + - ap y w) < eps_ u