let A be symmetrical Subset of COMPLEX; 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; ( F is_odd_on A implies F " is_odd_on A )
assume A1:
F is_odd_on A
; F " is_odd_on A
then A2:
A c= dom F
;
then A3:
A c= dom (F ")
by VALUED_1:def 7;
then A4:
dom ((F ") | A) = A
by RELAT_1:62;
A5:
F | A is odd
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 A3, A7, VALUED_1:def 7
.=
(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, Def6
.=
(- ((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 XCMPLX_1:222
.=
- ((F ") . x)
by A3, A6, VALUED_1:def 7
.=
- ((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_odd )
by A4;
hence
F " is_odd_on A
by A3; verum