const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const int : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const eps_ : set set term diadic_rational_p = \x:set.?y:set.y iIn omega & ?z:set.z iIn int & x = eps_ y * z const SNo : set prop axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) axiom omega_SNo: !x:set.x iIn omega -> SNo x const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const SNoS_ : set set axiom nonneg_diadic_rational_p_SNoS_omega: !x:set.x iIn omega -> !y:set.nat_p y -> eps_ x * y 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 axiom mul_SNo_minus_distrR: !x:set.!y:set.SNo x -> SNo y -> x * - y = - x * y lemma !x:set.!y:set.x iIn omega -> y iIn int -> (!z:set.z iIn omega -> eps_ x * z iIn SNoS_ omega) -> (!z:set.z iIn omega -> eps_ x * - z iIn SNoS_ omega) -> eps_ x * y iIn SNoS_ omega var x:set var y:set hyp x iIn omega hyp y iIn int claim (!z:set.z iIn omega -> eps_ x * z iIn SNoS_ omega) -> eps_ x * y iIn SNoS_ omega