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 F is_odd_on A ; :: thesis: - F is_odd_on A
then A1: ( A c= dom F & F | A is odd ) by Def8;
A2: A c= dom (- F) by A1, VALUED_1:8;
A3: dom ((- F) | A) = A by RELAT_1:91, A2;
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 A5: ( x in dom ((- F) | A) & - x in dom ((- F) | A) ) ; :: thesis: ((- F) | A) . (- x) = - (((- F) | A) . x)
then A6: ( x in dom (F | A) & - x in dom (F | A) ) by RELAT_1:91, A1, A3;
((- F) | A) . (- x) = ((- F) | A) /. (- x) by PARTFUN1:def 8, A5
.= (- F) /. (- x) by PARTFUN2:35, A2, A3, A5
.= (- F) . (- x) by PARTFUN1:def 8, A2, A3, A5
.= - (F . (- x)) by VALUED_1:8
.= - (F /. (- x)) by PARTFUN1:def 8, A3, A1, A5
.= - ((F | A) /. (- x)) by PARTFUN2:35, A1, A3, A5
.= - ((F | A) . (- x)) by PARTFUN1:def 8, A6
.= - (- ((F | A) . x)) by A1, Def6, A6
.= - (- ((F | A) /. x)) by PARTFUN1:def 8, A6
.= - (- (F /. x)) by PARTFUN2:35, A1, A3, A5
.= - (- (F . x)) by PARTFUN1:def 8, A1, A3, A5
.= - ((- F) . x) by VALUED_1:8
.= - ((- F) /. x) by PARTFUN1:def 8, A2, A3, A5
.= - (((- F) | A) /. x) by PARTFUN2:35, A2, A3, A5
.= - (((- F) | A) . x) by PARTFUN1:def 8, A5 ;
hence ((- F) | A) . (- x) = - (((- F) | A) . x) ; :: thesis: verum

end;

then ( (- F) | A is with_symmetrical_domain & (- F) | A is quasi_odd ) by Def6, A3, Def2;
hence - F is_odd_on A by A2, Def8; :: thesis: verum