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 |. F .| is even
proof
A1: dom F=dom(|. F .|) by VALUED_1:def 11;
  assume
A2: F is even;
  for x st x in dom(|. F .|) & -x in dom(|. F .|) holds (|. F .|).(-x)=(|.
  F .|).x
  proof
    let x;
    assume that
A3: x in dom(|. F .|) and
A4: -x in dom(|. F .|);
    (|. F .|).(-x)=|. F.(-x) .| by A4,VALUED_1:def 11
      .=|. F.x .| by A2,A1,A3,A4,Def3
      .=(|. F .|).x by A3,VALUED_1:def 11;
    hence thesis;
  end;
  then |. F .| is with_symmetrical_domain quasi_even by A2,A1;
  hence thesis;
end;
