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 omega : set const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_Le2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y <= z -> (x + y) <= x + z axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom add_SNo_minus_Le2b: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (z + y) <= x -> z <= x + - y axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom add_SNo_Lt1_cancel: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) < z + y -> x < z axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y axiom FalseE: ~ False axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x axiom add_SNo_minus_Lt1b: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < z + y -> (x + - y) < z const Empty : set const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x const SNoS_ : set set const ap : set set set const Sigma : set (set set) set const ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo (x + y) -> (!v:set.v iIn SNoS_ omega -> (!x2:set.x2 iIn omega -> abs_SNo (v + - y) < eps_ x2) -> v = y) -> (!v:set.v iIn omega -> SNo (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v)) -> (!v:set.v iIn omega -> (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v + - eps_ v) < x + y) -> SNo u -> y < u -> u iIn SNoS_ omega -> ~ (?v:set.v iIn omega & ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v <= x + u) -> Empty < u + - y -> u != y 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 + - y) < eps_ x2) -> v = y 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 y < 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 <= x + u claim ~ Empty < u + - y