const SNo_pair : set set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const CSNo_Re : set set const CSNo_Im : set set term add_CSNo = \x:set.\y:set.SNo_pair (CSNo_Re x + CSNo_Re y) (CSNo_Im x + CSNo_Im y) const In : set set prop term iIn = In infix iIn 2000 2000 const real : set const SNo : set prop axiom real_SNo: !x:set.x iIn real -> SNo x const Empty : set axiom add_SNo_0L: !x:set.SNo x -> Empty + x = x axiom add_SNo_0R: !x:set.SNo x -> x + Empty = x const mul_CSNo : set set set const Complex_i : set axiom real_Im_i_eq: !x:set.x iIn real -> CSNo_Im (mul_CSNo Complex_i x) = x axiom real_Re_i_eq: !x:set.x iIn real -> CSNo_Re (mul_CSNo Complex_i x) = Empty axiom real_Im_eq: !x:set.x iIn real -> CSNo_Im x = Empty axiom real_Re_eq: !x:set.x iIn real -> CSNo_Re x = x axiom complex_Im_eq: !x:set.x iIn real -> !y:set.y iIn real -> CSNo_Im (SNo_pair x y) = y axiom complex_Re_eq: !x:set.x iIn real -> !y:set.y iIn real -> CSNo_Re (SNo_pair x y) = x const complex : set axiom complex_E: !x:set.x iIn complex -> !P:prop.(!y:set.y iIn real -> !z:set.z iIn real -> x = SNo_pair y z -> P) -> P claim !x:set.x iIn complex -> x = add_CSNo (CSNo_Re x) (mul_CSNo Complex_i (CSNo_Im x))