let F be PartFunc of REAL,REAL; :: thesis: ( F is odd implies F " is odd )
A1: dom F = dom (F ") by VALUED_1:def 7;
assume A2: F is odd ; :: thesis: F " is odd
for x being Real st x in dom (F ") & - x in dom (F ") holds
(F ") . (- x) = - ((F ") . x)

proof

let x be Real; :: thesis: ( x in dom (F ") & - x in dom (F ") implies (F ") . (- x) = - ((F ") . x) )
assume that
A3: x in dom (F ") and
A4: - x in dom (F ") ; :: thesis: (F ") . (- x) = - ((F ") . x)
(F ") . (- x) = (F . (- x)) " by A4, VALUED_1:def 7
.= (- (F . x)) " by A2, A1, A3, A4, Def6
.= - ((F . x) ") by XCMPLX_1:222
.= - ((F ") . x) by A3, VALUED_1:def 7 ;
hence (F ") . (- x) = - ((F ") . x) ; :: thesis: verum

end;

then ( F " is with_symmetrical_domain & F " is quasi_odd ) by A2, A1;
hence F " is odd ; :: thesis: verum