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 axiom SNoLt_irref: !x:set.~ x < x const omega : set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const ordsucc : set set const Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const minus_SNo : set set term - = minus_SNo const eps_ : set set const SNoS_ : set set const abs_SNo : set set var x:set var y:set hyp SNo (x * y) hyp !z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x * y) < eps_ w) -> z = x * y hyp ordsucc Empty < x * y hyp ~ ?z:set.z iIn omega & ordsucc Empty <= x * y + - eps_ z claim ordsucc Empty != x * y