const In : set set prop term iIn = In infix iIn 2000 2000 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 Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x axiom SNoLt_irref: !x:set.~ x < x const Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const omega : set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const ap : set set set const Sigma : set (set set) set const ordsucc : set set const SNoS_ : set set const eps_ : set set const abs_SNo : set set var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp SNo (x + y) hyp !v:set.v iIn SNoS_ omega -> (!x2:set.x2 iIn omega -> abs_SNo (v + - x) < eps_ x2) -> v = x hyp !v:set.v iIn omega -> SNo (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v) hyp !v:set.v iIn omega -> (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v + - eps_ v) < x + y hyp SNo u hyp x < u hyp u iIn SNoS_ omega hyp ~ ?v:set.v iIn omega & ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v <= u + y hyp Empty < u + - x claim u != x