const CSNo : set prop const Complex_i : set axiom CSNo_Complex_i: CSNo Complex_i const ordsucc : set set const Empty : set axiom CSNo_1: CSNo (ordsucc Empty) const SNo : set prop axiom SNo_0: SNo Empty axiom SNo_1: SNo (ordsucc Empty) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_0L: !x:set.SNo x -> Empty + x = x const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom mul_SNo_oneL: !x:set.SNo x -> ordsucc Empty * x = x axiom mul_SNo_zeroL: !x:set.SNo x -> Empty * x = Empty const CSNo_Re : set set axiom Re_1: CSNo_Re (ordsucc Empty) = ordsucc Empty const CSNo_Im : set set axiom Im_i: CSNo_Im Complex_i = ordsucc Empty axiom Re_i: CSNo_Re Complex_i = Empty const minus_CSNo : set set axiom minus_CSNo_CRe: !x:set.CSNo x -> CSNo_Re (minus_CSNo x) = - CSNo_Re x const mul_CSNo : set 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 axiom mul_SNo_zeroR: !x:set.SNo x -> x * Empty = Empty axiom minus_SNo_0: - Empty = Empty axiom Im_1: CSNo_Im (ordsucc Empty) = Empty axiom minus_CSNo_CIm: !x:set.CSNo x -> CSNo_Im (minus_CSNo x) = - CSNo_Im 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 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 hyp CSNo (mul_CSNo Complex_i Complex_i) claim CSNo (minus_CSNo (ordsucc Empty)) -> mul_CSNo Complex_i Complex_i = minus_CSNo (ordsucc Empty)