const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo axiom abs_SNo_dist_swap: !x:set.!y:set.SNo x -> SNo y -> abs_SNo (x + - y) = abs_SNo (y + - x) const omega : set const ap : set set set const SNoS_ : set set const eps_ : set set const ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.SNo x -> (!u:set.u iIn omega -> SNo (ap y u)) -> z iIn SNoS_ omega -> (!u:set.u iIn omega -> abs_SNo (z + - x) < eps_ u) -> SNo z -> SNo - z -> SNo (x + - z) -> w iIn omega -> (!u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - ap y (ordsucc w)) < eps_ v) -> u = ap y (ordsucc w)) -> SNo (ap y (ordsucc w)) -> z < ap y (ordsucc w) -> Empty < ap y (ordsucc w) + - z -> SNo (ap y (ordsucc w) + - z) -> ap y (ordsucc w) < x -> Empty < x + - z -> abs_SNo (z + - x) = x + - z -> ap y w < z var x:set var y:set var z:set var w:set hyp SNo x hyp !u:set.u iIn omega -> SNo (ap y u) hyp z iIn SNoS_ omega hyp !u:set.u iIn omega -> abs_SNo (z + - x) < eps_ u hyp SNo z hyp SNo - z hyp SNo (x + - z) hyp w iIn omega hyp !u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - ap y (ordsucc w)) < eps_ v) -> u = ap y (ordsucc w) hyp SNo (ap y (ordsucc w)) hyp z < ap y (ordsucc w) hyp Empty < ap y (ordsucc w) + - z hyp SNo (ap y (ordsucc w) + - z) hyp ap y (ordsucc w) < x hyp z < x claim Empty < x + - z -> ap y w < z