let A be symmetrical Subset of COMPLEX; 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; ( F is_even_on A implies - F is_even_on A )
assume A1:
F is_even_on A
; - 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;
( 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)
;
((- 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
;
verum
end;
then
( (- F) | A is with_symmetrical_domain & (- F) | A is quasi_even )
by A4;
hence
- F is_even_on A
by A3; verum