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 2;
  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 2
      .=r+F.x by A2,A1,A3,A4,Def3
      .=(r+F).x by A3,VALUED_1:def 2;
    hence thesis;
  end;
  then r+F is with_symmetrical_domain quasi_even by A2,A1;
  hence thesis;
end;
