const CSNo : set prop const Complex_i : set axiom CSNo_Complex_i: CSNo Complex_i const mul_CSNo : set set set axiom CSNo_mul_CSNo: !x:set.!y:set.CSNo x -> CSNo y -> CSNo (mul_CSNo x y) const SNo : set prop 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 CSNo_Im : set set const ordsucc : set set const Empty : set const add_CSNo : set set set const minus_CSNo : set 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) -> SNo (y * CSNo_Re x) -> CSNo (y * CSNo_Re x) -> SNo (y * CSNo_Im x) -> CSNo (y * CSNo_Im x) -> CSNo (mul_CSNo Complex_i (y * CSNo_Im x)) -> mul_CSNo x (add_CSNo (y * CSNo_Re x) (minus_CSNo (mul_CSNo Complex_i (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 hyp SNo (CSNo_Re x) hyp SNo (CSNo_Im x) hyp SNo (y * CSNo_Re x) hyp CSNo (y * CSNo_Re x) hyp SNo (y * CSNo_Im x) claim CSNo (y * CSNo_Im x) -> mul_CSNo x (add_CSNo (y * CSNo_Re x) (minus_CSNo (mul_CSNo Complex_i (y * CSNo_Im x)))) = ordsucc Empty