let F be PartFunc of REAL,REAL; :: thesis: ( F is even implies F ^2 is even )
A1: dom F = dom (F ^2) by VALUED_1:11;
assume A2: F is even ; :: thesis: F ^2 is even
for x being Real st x in dom (F ^2) & - x in dom (F ^2) holds
(F ^2) . (- x) = (F ^2) . x

proof

let x be Real; :: thesis: ( x in dom (F ^2) & - x in dom (F ^2) implies (F ^2) . (- x) = (F ^2) . x )
assume A3: ( x in dom (F ^2) & - x in dom (F ^2) ) ; :: thesis: (F ^2) . (- x) = (F ^2) . x
(F ^2) . (- x) = (F . (- x)) ^2 by VALUED_1:11
.= (F . x) ^2 by A2, A1, A3, Def3
.= (F ^2) . x by VALUED_1:11 ;
hence (F ^2) . (- x) = (F ^2) . x ; :: thesis: verum

end;

then ( F ^2 is with_symmetrical_domain & F ^2 is quasi_even ) by A2, A1;
hence F ^2 is even ; :: thesis: verum