let F, G be PartFunc of REAL ,REAL ; :: thesis: ( F is odd & G is odd & (dom F) /\ (dom G) is symmetrical implies F /" G is even )
assume that
A1: F is odd and
A2: G is odd and
A3: (dom F) /\ (dom G) is symmetrical ; :: thesis: F /" G is even
A7: (dom F) /\ (dom G) = dom (F /" G) by VALUED_1:16;
then A8: ( dom (F /" G) c= dom F & dom (F /" G) c= dom G ) by XBOOLE_1:17;
for x being Real st x in dom (F /" G) & - x in dom (F /" G) holds
(F /" G) . (- x) = (F /" G) . x

proof

let x be Real; :: thesis: ( x in dom (F /" G) & - x in dom (F /" G) implies (F /" G) . (- x) = (F /" G) . x )
assume A9: ( x in dom (F /" G) & - x in dom (F /" G) ) ; :: thesis: (F /" G) . (- x) = (F /" G) . x
(F /" G) . (- x) = (F . (- x)) / (G . (- x)) by VALUED_1:17
.= (- (F . x)) / (G . (- x)) by A1, Def6, A8, A9
.= (- (F . x)) / (- (G . x)) by A2, Def6, A8, A9
.= (F . x) / (G . x) by XCMPLX_1:192
.= (F /" G) . x by VALUED_1:17 ;
hence (F /" G) . (- x) = (F /" G) . x ; :: thesis: verum

end;

then ( F /" G is with_symmetrical_domain & F /" G is quasi_even ) by Def3, A3, A7, Def2;
hence F /" G is even ; :: thesis: verum