reserve x, r for Real;
reserve A for symmetrical Subset of COMPLEX;
reserve F,G for PartFunc of REAL, REAL;

theorem
  F is even implies r (#) F is even
proof
A1: dom F=dom(r (#) F) by VALUED_1:def 5;
  assume
A2: F is even;
  for x st x in dom(r (#) F) & -x in dom(r (#) F) holds (r (#) F).(-x)=(r
  (#) F).x
  proof
    let x;
    assume that
A3: x in dom(r (#) F) and
A4: -x in dom(r (#) F);
    (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 thesis;
  end;
  then r (#) F is with_symmetrical_domain quasi_even by A2,A1;
  hence thesis;
end;
