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
A7: (dom F) /\ (dom G) = dom (F - G) by VALUED_1:12;
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_1:13, A9
.= (- (F . x)) - (G . (- x)) by A1, Def6, A8, A9
.= (- (F . x)) - (- (G . x)) by A2, Def6, A8, A9
.= - ((F . x) - (G . x))
.= - ((F - G) . x) by VALUED_1:13, A9 ;
hence (F - G) . (- x) = - ((F - G) . x) ; :: thesis: verum

end;

then ( F - G is with_symmetrical_domain & F - G is quasi_odd ) by Def6, A3, A7, Def2;
hence F - G is odd ; :: thesis: verum