const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const real : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom real_mul_SNo: !x:set.x iIn real -> !y:set.y iIn real -> x * y iIn real const omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const ordsucc : set set axiom pos_small_real_recip_ex: !x:set.x iIn real -> Empty < x -> x < ordsucc Empty -> ?y:set.y iIn real & x * y = ordsucc Empty lemma !x:set.!y:set.!z:set.SNo x -> y iIn omega -> z iIn real -> (eps_ y * x) * z = ordsucc Empty -> SNo (eps_ y) -> ?w:set.w iIn real & x * w = ordsucc Empty var x:set var y:set hyp x iIn real hyp Empty < x hyp SNo x hyp y iIn omega hyp eps_ y * x < ordsucc Empty hyp eps_ y iIn real claim Empty < eps_ y * x -> ?z:set.z iIn real & x * z = ordsucc Empty