let A be symmetrical Subset of COMPLEX ; :: thesis: for F being PartFunc of REAL ,REAL st F is_odd_on A holds
F " is_odd_on A
let F be PartFunc of REAL ,REAL ; :: thesis: ( F is_odd_on A implies F " is_odd_on A )
assume A1: F is_odd_on A ; :: thesis: F " is_odd_on A
then A2: A c= dom F by Def8;
then A3: A c= dom (F " ) by VALUED_1:def 7;
then A4: dom ((F " ) | A) = A by RELAT_1:91;
A5: F | A is odd by A1, Def8;
for x being Real st x in dom ((F " ) | A) & - x in dom ((F " ) | A) holds
((F " ) | A) . (- x) = - (((F " ) | A) . x)

proof

let x be Real; :: thesis: ( x in dom ((F " ) | A) & - x in dom ((F " ) | A) implies ((F " ) | A) . (- x) = - (((F " ) | A) . x) )
assume that
A6: x in dom ((F " ) | A) and
A7: - x in dom ((F " ) | A) ; :: thesis: ((F " ) | A) . (- x) = - (((F " ) | A) . x)
A8: x in dom (F | A) by A2, A4, A6, RELAT_1:91;
A9: - x in dom (F | A) by A2, A4, A7, RELAT_1:91;
((F " ) | A) . (- x) = ((F " ) | A) /. (- x) by A7, PARTFUN1:def 8
.= (F " ) /. (- x) by A3, A4, A7, PARTFUN2:35
.= (F " ) . (- x) by A3, A4, A7, PARTFUN1:def 8
.= (F . (- x)) " by A3, A4, A7, VALUED_1:def 7
.= (F /. (- x)) " by A2, A4, A7, PARTFUN1:def 8
.= ((F | A) /. (- x)) " by A2, A4, A7, PARTFUN2:35
.= ((F | A) . (- x)) " by A9, PARTFUN1:def 8
.= (- ((F | A) . x)) " by A5, A8, A9, Def6
.= (- ((F | A) /. x)) " by A8, PARTFUN1:def 8
.= (- (F /. x)) " by A2, A4, A6, PARTFUN2:35
.= (- (F . x)) " by A2, A4, A6, PARTFUN1:def 8
.= - ((F . x) " ) by XCMPLX_1:224
.= - ((F " ) . x) by A3, A4, A6, VALUED_1:def 7
.= - ((F " ) /. x) by A3, A4, A6, PARTFUN1:def 8
.= - (((F " ) | A) /. x) by A3, A4, A6, PARTFUN2:35
.= - (((F " ) | A) . x) by A6, PARTFUN1:def 8 ;
hence ((F " ) | A) . (- x) = - (((F " ) | A) . x) ; :: thesis: verum

end;

then ( (F " ) | A is with_symmetrical_domain & (F " ) | A is quasi_odd ) by A4, Def2, Def6;
hence F " is_odd_on A by A3, Def8; :: thesis: verum