let A be symmetrical Subset of COMPLEX; :: thesis: for F being PartFunc of REAL,REAL st A c= dom F & ( for x being Real st x in A holds

(F . x) / (F . (- x)) = - 1 ) holds

F is_odd_on A

let F be PartFunc of REAL,REAL; :: thesis: ( A c= dom F & ( for x being Real st x in A holds

(F . x) / (F . (- x)) = - 1 ) implies F is_odd_on A )

assume that

A1: A c= dom F and

A2: for x being Real st x in A holds

(F . x) / (F . (- x)) = - 1 ; :: thesis: F is_odd_on A

A3: dom (F | A) = A by A1, RELAT_1:62;

A4: for x being Real st x in A holds

F . (- x) = - (F . x)

(F | A) . (- x) = - ((F | A) . x)

hence F is_odd_on A by A1; :: thesis: verum

(F . x) / (F . (- x)) = - 1 ) holds

F is_odd_on A

let F be PartFunc of REAL,REAL; :: thesis: ( A c= dom F & ( for x being Real st x in A holds

(F . x) / (F . (- x)) = - 1 ) implies F is_odd_on A )

assume that

A1: A c= dom F and

A2: for x being Real st x in A holds

(F . x) / (F . (- x)) = - 1 ; :: thesis: F is_odd_on A

A3: dom (F | A) = A by A1, RELAT_1:62;

A4: for x being Real st x in A holds

F . (- x) = - (F . x)

proof

for x being Real st x in dom (F | A) & - x in dom (F | A) holds
let x be Real; :: thesis: ( x in A implies F . (- x) = - (F . x) )

assume x in A ; :: thesis: F . (- x) = - (F . x)

then (F . x) / (F . (- x)) = - 1 by A2;

hence F . (- x) = - (F . x) by XCMPLX_1:195; :: thesis: verum

end;assume x in A ; :: thesis: F . (- x) = - (F . x)

then (F . x) / (F . (- x)) = - 1 by A2;

hence F . (- x) = - (F . x) by XCMPLX_1:195; :: thesis: verum

(F | A) . (- x) = - ((F | A) . x)

proof

then
( F | A is with_symmetrical_domain & F | A is quasi_odd )
by A3;
let x be Real; :: thesis: ( x in dom (F | A) & - x in dom (F | A) implies (F | A) . (- x) = - ((F | A) . x) )

assume that

A5: x in dom (F | A) and

A6: - x in dom (F | A) ; :: thesis: (F | A) . (- x) = - ((F | A) . x)

reconsider x = x as Element of REAL by XREAL_0:def 1;

(F | A) . (- x) = (F | A) /. (- x) by A6, PARTFUN1:def 6

.= F /. (- x) by A1, A3, A6, PARTFUN2:17

.= F . (- x) by A1, A6, PARTFUN1:def 6

.= - (F . x) by A4, A5

.= - (F /. x) by A1, A5, PARTFUN1:def 6

.= - ((F | A) /. x) by A1, A3, A5, PARTFUN2:17

.= - ((F | A) . x) by A5, PARTFUN1:def 6 ;

hence (F | A) . (- x) = - ((F | A) . x) ; :: thesis: verum

end;assume that

A5: x in dom (F | A) and

A6: - x in dom (F | A) ; :: thesis: (F | A) . (- x) = - ((F | A) . x)

reconsider x = x as Element of REAL by XREAL_0:def 1;

(F | A) . (- x) = (F | A) /. (- x) by A6, PARTFUN1:def 6

.= F /. (- x) by A1, A3, A6, PARTFUN2:17

.= F . (- x) by A1, A6, PARTFUN1:def 6

.= - (F . x) by A4, A5

.= - (F /. x) by A1, A5, PARTFUN1:def 6

.= - ((F | A) /. x) by A1, A3, A5, PARTFUN2:17

.= - ((F | A) . x) by A5, PARTFUN1:def 6 ;

hence (F | A) . (- x) = - ((F | A) . x) ; :: thesis: verum

hence F is_odd_on A by A1; :: thesis: verum