let A be symmetrical Subset of COMPLEX; :: thesis: for F being PartFunc of REAL,REAL st F is_even_on A holds
- F is_even_on A
let F be PartFunc of REAL,REAL; :: thesis: ( F is_even_on A implies - F is_even_on A )
assume A1: F is_even_on A ; :: thesis: - F is_even_on A
then A2: A c= dom F ;
then A3: A c= dom (- F) by VALUED_1:8;
then A4: dom ((- F) | A) = A by RELAT_1:62;
A5: F | A is even by A1;
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_1:62;
A9: - x in dom (F | A) by A2, A4, A7, RELAT_1:62;
reconsider x = x as Element of REAL by XREAL_0:def 1;
((- F) | A) . (- x) = ((- F) | A) /. (- x) by A7, PARTFUN1:def 6
.= (- F) /. (- x) by A3, A4, A7, PARTFUN2:17
.= (- F) . (- x) by A3, A7, PARTFUN1:def 6
.= - (F . (- x)) by VALUED_1:8
.= - (F /. (- x)) by A2, A7, PARTFUN1:def 6
.= - ((F | A) /. (- x)) by A2, A4, A7, PARTFUN2:17
.= - ((F | A) . (- x)) by A9, PARTFUN1:def 6
.= - ((F | A) . x) by A5, A8, A9, Def3
.= - ((F | A) /. x) by A8, PARTFUN1:def 6
.= - (F /. x) by A2, A4, A6, PARTFUN2:17
.= - (F . x) by A2, A6, PARTFUN1:def 6
.= (- F) . x by VALUED_1:8
.= (- F) /. x by A3, A6, PARTFUN1:def 6
.= ((- F) | A) /. x by A3, A4, A6, PARTFUN2:17
.= ((- F) | A) . x by A6, PARTFUN1:def 6 ;
hence ((- F) | A) . (- x) = ((- F) | A) . x ; :: thesis: verum

end;

then ( (- F) | A is with_symmetrical_domain & (- F) | A is quasi_even ) by A4;
hence - F is_even_on A by A3; :: thesis: verum