let r be Real; :: thesis: for F being PartFunc of REAL,REAL st F is even holds
F - r is even
let F be PartFunc of REAL,REAL; :: thesis: ( F is even implies F - r is even )
A1: dom F = dom (F - r) by VALUED_1:3;
assume A2: F is even ; :: thesis: F - r is even
for x being Real st x in dom (F - r) & - x in dom (F - r) holds
(F - r) . (- x) = (F - r) . x

proof

let x be Real; :: thesis: ( x in dom (F - r) & - x in dom (F - r) implies (F - r) . (- x) = (F - r) . x )
assume that
A3: x in dom (F - r) and
A4: - x in dom (F - r) ; :: thesis: (F - r) . (- x) = (F - r) . x
A5: x in dom F by A3, VALUED_1:3;
- x in dom F by A4, VALUED_1:3;
then (F - r) . (- x) = (F . (- x)) - r by VALUED_1:3
.= (F . x) - r by A2, A1, A3, A4, Def3
.= (F - r) . x by A5, VALUED_1:3 ;
hence (F - r) . (- x) = (F - r) . x ; :: thesis: verum

end;

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