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 minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const omega : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const ap : set set set const eps_ : set set const ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.(!v:set.v iIn omega -> (ap z v + - eps_ v) < x) -> (!v:set.v iIn omega -> (ap w v + - eps_ v) < y) -> SNo x -> SNo y -> (!v:set.v iIn omega -> SNo (ap z (ordsucc v))) -> (!v:set.v iIn omega -> SNo (ap w (ordsucc v))) -> u iIn omega -> SNo (eps_ (ordsucc u)) -> SNo - eps_ (ordsucc u) -> ((ap z (ordsucc u) + ap w (ordsucc u)) + - (eps_ (ordsucc u) + eps_ (ordsucc u))) < x + y var x:set var y:set var z:set var w:set var u:set hyp !v:set.v iIn omega -> (ap z v + - eps_ v) < x hyp !v:set.v iIn omega -> (ap w v + - eps_ v) < y hyp SNo x hyp SNo y hyp !v:set.v iIn omega -> SNo (ap z (ordsucc v)) hyp !v:set.v iIn omega -> SNo (ap w (ordsucc v)) hyp u iIn omega claim SNo (eps_ (ordsucc u)) -> ((ap z (ordsucc u) + ap w (ordsucc u)) + - (eps_ (ordsucc u) + eps_ (ordsucc u))) < x + y