const CSNo : set prop const CSNo_Re : set set const mul_CSNo : set set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const minus_SNo : set set term - = minus_SNo const CSNo_Im : set set axiom mul_CSNo_CRe: !x:set.!y:set.CSNo x -> CSNo y -> CSNo_Re (mul_CSNo x y) = CSNo_Re x * CSNo_Re y + - CSNo_Im x * CSNo_Im y const SNo : set prop axiom mul_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x * y = y * x axiom mul_CSNo_CIm: !x:set.!y:set.CSNo x -> CSNo y -> CSNo_Im (mul_CSNo x y) = CSNo_Re x * CSNo_Im y + CSNo_Im x * CSNo_Re y axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom CSNo_ReIm_split: !x:set.!y:set.CSNo x -> CSNo y -> CSNo_Re x = CSNo_Re y -> CSNo_Im x = CSNo_Im y -> x = y var x:set var y:set hyp CSNo x hyp CSNo y hyp CSNo (mul_CSNo x y) hyp CSNo (mul_CSNo y x) hyp SNo (CSNo_Re x) hyp SNo (CSNo_Re y) hyp SNo (CSNo_Im x) claim SNo (CSNo_Im y) -> mul_CSNo x y = mul_CSNo y x