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 omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom minus_SNo_Lt_contra: !x:set.!y:set.SNo x -> SNo y -> x < y -> - y < - x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < 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 ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo - x -> z iIn SNoS_ omega -> (!u:set.u iIn omega -> abs_SNo (z + - x) < eps_ u) -> SNo z -> SNo (z + - x) -> (!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)) -> ap y (ordsucc w) < z -> Empty < z + - ap y (ordsucc w) -> SNo (z + - ap y (ordsucc w)) -> x < ap y (ordsucc w) -> abs_SNo (z + - x) = z + - x -> z = ap y (ordsucc w) -> z < ap y w var x:set var y:set var z:set var w:set hyp SNo x hyp SNo - x hyp z iIn SNoS_ omega hyp !u:set.u iIn omega -> abs_SNo (z + - x) < eps_ u hyp SNo z hyp SNo (z + - x) 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 ap y (ordsucc w) < z hyp Empty < z + - ap y (ordsucc w) hyp SNo (z + - ap y (ordsucc w)) hyp x < ap y (ordsucc w) hyp Empty < z + - x claim abs_SNo (z + - x) = z + - x -> z < ap y w