const CSNo : set prop const SNo : set prop const CSNo_Im : set set axiom CSNo_ImR: !x:set.CSNo x -> SNo (CSNo_Im x) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const CSNo_Re : set set const ordsucc : set set const Empty : set const mul_CSNo : set set set const add_CSNo : set set set const minus_CSNo : set set const Complex_i : set lemma !x:set.!y:set.CSNo x -> SNo y -> (CSNo_Re x * CSNo_Re x + CSNo_Im x * CSNo_Im x) * y = ordsucc Empty -> SNo (CSNo_Re x) -> SNo (CSNo_Im x) -> mul_CSNo x (add_CSNo (mul_CSNo y (CSNo_Re x)) (minus_CSNo (mul_CSNo Complex_i (mul_CSNo y (CSNo_Im x))))) = ordsucc Empty var x:set var y:set hyp CSNo x hyp SNo y hyp (CSNo_Re x * CSNo_Re x + CSNo_Im x * CSNo_Im x) * y = ordsucc Empty claim SNo (CSNo_Re x) -> mul_CSNo x (add_CSNo (mul_CSNo y (CSNo_Re x)) (minus_CSNo (mul_CSNo Complex_i (mul_CSNo y (CSNo_Im x))))) = ordsucc Empty