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 odd )
assume that
A1: F is odd and
A2: G is odd and
A3: (dom F) /\ (dom G) is symmetrical ; :: thesis: F + G is odd
A4: (dom F) /\ (dom G) = dom (F + G) by VALUED_1:def 1;
then A5: dom (F + G) c= dom G by XBOOLE_1:17;
A6: dom (F + G) c= dom F by A4, 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 that
A7: x in dom (F + G) and
A8: - x in dom (F + G) ; :: thesis: (F + G) . (- x) = - ((F + G) . x)
(F + G) . (- x) = (F . (- x)) + (G . (- x)) by A8, VALUED_1:def 1
.= (- (F . x)) + (G . (- x)) by A1, A6, A7, A8, Def6
.= (- (F . x)) + (- (G . x)) by A2, A5, A7, A8, Def6
.= - ((F . x) + (G . x))
.= - ((F + G) . x) by A7, VALUED_1:def 1 ;
hence (F + G) . (- x) = - ((F + G) . x) ; :: thesis: verum

end;

then ( F + G is with_symmetrical_domain & F + G is quasi_odd ) by A3, A4;
hence F + G is odd ; :: thesis: verum