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 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.z iIn omega -> SNo (eps_ z) -> w iIn int -> x = eps_ z * w -> u iIn omega -> SNo (eps_ u) -> v iIn int -> y = eps_ u * v -> SNo v -> SNo (eps_ u * v) -> ?x2:set.x2 iIn omega & ?y2:set.y2 iIn int & x + y = eps_ x2 * y2 var x:set var y:set var z:set var w:set var u:set var v:set hyp z iIn omega hyp SNo (eps_ z) hyp w iIn int hyp x = eps_ z * w hyp u iIn omega hyp SNo (eps_ u) hyp v iIn int hyp y = eps_ u * v claim SNo v -> ?x2:set.x2 iIn omega & ?y2:set.y2 iIn int & x + y = eps_ x2 * y2