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 omega : set const ap : set set set const SNoCut : set set set const Sep : set (set prop) set const SNoS_ : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 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 Empty : set lemma !x:set.!y:set.!z:set.!w:set.!u:set.(!v:set.v iIn omega -> SNo (ap x v)) -> 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)) -> (!v:set.v iIn omega -> ap x v <= 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)) -> (!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) -> SNo z -> w iIn omega -> z < ap x w -> u iIn omega -> Empty < ap x w + - z -> ap x w <= 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 x w) < eps_ u var x:set var y:set var z:set var w:set var u:set hyp !v:set.v iIn omega -> SNo (ap x 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 -> ap x v <= 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) 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 z < ap x w hyp u iIn omega claim Empty < ap x w + - z -> abs_SNo (z + - ap x w) < eps_ u