let F be PartFunc of REAL,REAL; :: thesis: ( F is odd implies |.F.| is even )
A1: dom F = dom |.F.| by VALUED_1:def 11;
assume A2: F is odd ; :: thesis: |.F.| is even
for x being Real st x in dom |.F.| & - x in dom |.F.| holds
|.F.| . (- x) = |.F.| . x

proof

let x be Real; :: thesis: ( x in dom |.F.| & - x in dom |.F.| implies |.F.| . (- x) = |.F.| . x )
assume that
A3: x in dom |.F.| and
A4: - x in dom |.F.| ; :: thesis: |.F.| . (- x) = |.F.| . x
|.F.| . (- x) = |.(F . (- x)).| by A4, VALUED_1:def 11
.= |.(- (F . x)).| by A2, A1, A3, A4, Def6
.= |.(F . x).| by COMPLEX1:52
.= |.F.| . x by A3, VALUED_1:def 11 ;
hence |.F.| . (- x) = |.F.| . x ; :: thesis: verum

end;

then ( |.F.| is with_symmetrical_domain & |.F.| is quasi_even ) by A2, A1;
hence |.F.| is even ; :: thesis: verum