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^2 is even
proof
A1: dom F=dom(F^2) by VALUED_1:11;
  assume
A2: F is even;
  for x st x in dom(F^2) & -x in dom(F^2) holds (F^2).(-x)=(F^2).x
  proof
    let x;
    assume
A3: x in dom(F^2) & -x in dom(F^2);
    (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 thesis;
  end;
  then F^2 is with_symmetrical_domain quasi_even by A2,A1;
  hence thesis;
end;
