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 add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_In_omega: !x:set.x iIn omega -> !y:set.y iIn omega -> x + y iIn omega axiom int_mul_SNo: !x:set.x iIn int -> !y:set.y iIn int -> x * y iIn int const SNo : set prop axiom mul_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 mul_SNo_eps_eps_add_SNo: !x:set.x iIn omega -> !y:set.y iIn omega -> eps_ x * eps_ y = eps_ (x + y) 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 SNo w hyp u iIn omega hyp SNo (eps_ u) hyp v iIn int hyp SNo v claim SNo (eps_ u * v) -> y = eps_ u * v -> ?x2:set.x2 iIn omega & ?y2:set.y2 iIn int & x * y = eps_ x2 * y2