const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const eps_ : set set const SNoS_ : set set axiom SNo_eps_SNoS_omega: !x:set.x iIn omega -> eps_ x iIn SNoS_ omega const minus_SNo : set set term - = minus_SNo axiom minus_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> - x iIn SNoS_ omega const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> !y:set.y iIn SNoS_ omega -> x + y iIn SNoS_ omega const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom add_SNo_eps_Lt: !x:set.SNo x -> !y:set.y iIn omega -> x < x + eps_ y axiom add_SNo_minus_Lt1b: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < z + y -> (x + - y) < z const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const Empty : set axiom add_SNo_0R: !x:set.SNo x -> x + Empty = x axiom add_SNo_minus_SNo_linv: !x:set.SNo x -> - x + x = Empty axiom add_SNo_com_4_inner_mid: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (x + y) + z + w = (x + z) + y + w axiom eps_ordsucc_half_add: !x:set.nat_p x -> eps_ (ordsucc x) + eps_ (ordsucc x) = eps_ x var x:set var y:set hyp x iIn SNoS_ omega hyp SNo x hyp y iIn omega claim SNo (eps_ (ordsucc y)) -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)