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.z iIn omega -> SNo (eps_ z) -> w iIn int -> x = eps_ z * w -> SNo w -> SNo (eps_ z * w) -> diadic_rational_p y -> diadic_rational_p (x + y) var x:set var y:set var z:set var w:set hyp z iIn omega hyp SNo (eps_ z) hyp w iIn int hyp x = eps_ z * w claim SNo w -> diadic_rational_p y -> diadic_rational_p (x + y)