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 const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const SNoS_ : set set 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 const real : set const ap : set set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const minus_SNo : set set term - = minus_SNo const abs_SNo : set set const ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.x iIn real -> y iIn real -> (!x2:set.x2 iIn omega -> ap z x2 < x) -> (!x2:set.x2 iIn omega -> x < ap z x2 + eps_ x2) -> (!x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap z y2 < ap z x2) -> (!x2:set.x2 iIn omega -> (ap w x2 + - eps_ x2) < x) -> (!x2:set.x2 iIn omega -> x < ap w x2) -> (!x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap w x2 < ap w y2) -> (!x2:set.x2 iIn omega -> ap u x2 < y) -> (!x2:set.x2 iIn omega -> y < ap u x2 + eps_ x2) -> (!x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap u y2 < ap u x2) -> (!x2:set.x2 iIn omega -> (ap v x2 + - eps_ x2) < y) -> (!x2:set.x2 iIn omega -> y < ap v x2) -> (!x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap v x2 < ap v y2) -> SNo x -> SNo y -> SNo (x + y) -> (!x2:set.x2 iIn SNoS_ omega -> (!y2:set.y2 iIn omega -> abs_SNo (x2 + - x) < eps_ y2) -> x2 = x) -> (!x2:set.x2 iIn SNoS_ omega -> (!y2:set.y2 iIn omega -> abs_SNo (x2 + - y) < eps_ y2) -> x2 = y) -> (!x2:set.x2 iIn omega -> ap z (ordsucc x2) iIn SNoS_ omega) -> (!x2:set.x2 iIn omega -> SNo (ap z (ordsucc x2))) -> (!x2:set.x2 iIn omega -> ap u (ordsucc x2) iIn SNoS_ omega) -> (!x2:set.x2 iIn omega -> SNo (ap u (ordsucc x2))) -> (!x2:set.x2 iIn omega -> ap w (ordsucc x2) iIn SNoS_ omega) -> (!x2:set.x2 iIn omega -> SNo (ap w (ordsucc x2))) -> (!x2:set.x2 iIn omega -> ap v (ordsucc x2) iIn SNoS_ omega) -> (!x2:set.x2 iIn omega -> SNo (ap v (ordsucc x2))) -> x + y iIn real var x:set var y:set var z:set var w:set var u:set var v:set hyp x iIn real hyp y iIn real hyp !x2:set.x2 iIn omega -> ap z x2 < x hyp !x2:set.x2 iIn omega -> x < ap z x2 + eps_ x2 hyp !x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap z y2 < ap z x2 hyp !x2:set.x2 iIn omega -> (ap w x2 + - eps_ x2) < x hyp !x2:set.x2 iIn omega -> x < ap w x2 hyp !x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap w x2 < ap w y2 hyp v iIn setexp (SNoS_ omega) omega hyp !x2:set.x2 iIn omega -> ap u x2 < y hyp !x2:set.x2 iIn omega -> y < ap u x2 + eps_ x2 hyp !x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap u y2 < ap u x2 hyp !x2:set.x2 iIn omega -> (ap v x2 + - eps_ x2) < y hyp !x2:set.x2 iIn omega -> y < ap v x2 hyp !x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap v x2 < ap v y2 hyp SNo x hyp SNo y hyp SNo (x + y) hyp !x2:set.x2 iIn SNoS_ omega -> (!y2:set.y2 iIn omega -> abs_SNo (x2 + - x) < eps_ y2) -> x2 = x hyp !x2:set.x2 iIn SNoS_ omega -> (!y2:set.y2 iIn omega -> abs_SNo (x2 + - y) < eps_ y2) -> x2 = y hyp !x2:set.x2 iIn omega -> ap z (ordsucc x2) iIn SNoS_ omega hyp !x2:set.x2 iIn omega -> SNo (ap z (ordsucc x2)) hyp !x2:set.x2 iIn omega -> ap u (ordsucc x2) iIn SNoS_ omega hyp !x2:set.x2 iIn omega -> SNo (ap u (ordsucc x2)) hyp !x2:set.x2 iIn omega -> ap w (ordsucc x2) iIn SNoS_ omega hyp !x2:set.x2 iIn omega -> SNo (ap w (ordsucc x2)) claim (!x2:set.x2 iIn omega -> ap v (ordsucc x2) iIn SNoS_ omega) -> x + y iIn real