let r be Real; :: thesis: for F being PartFunc of REAL,REAL st F is even holds
r (#) F is even
let F be PartFunc of REAL,REAL; :: thesis: ( F is even implies r (#) F is even )
A1: dom F = dom (r (#) F) by VALUED_1:def 5;
assume A2: F is even ; :: thesis: r (#) F is even
for x being Real st x in dom (r (#) F) & - x in dom (r (#) F) holds
(r (#) F) . (- x) = (r (#) F) . x

proof

let x be Real; :: thesis: ( x in dom (r (#) F) & - x in dom (r (#) F) implies (r (#) F) . (- x) = (r (#) F) . x )
assume that
A3: x in dom (r (#) F) and
A4: - x in dom (r (#) F) ; :: thesis: (r (#) F) . (- x) = (r (#) F) . x
(r (#) F) . (- x) = r * (F . (- x)) by A4, VALUED_1:def 5
.= r * (F . x) by A2, A1, A3, A4, Def3
.= (r (#) F) . x by A3, VALUED_1:def 5 ;
hence (r (#) F) . (- x) = (r (#) F) . x ; :: thesis: verum

end;

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