theorem
  F is idempotent implies F.:(f,f) = f
proof
  assume
A1: F is idempotent;
  per cases;
  suppose
A2: Y = {};
    hence F.:(f,f) = {}
      .= f by A2;
  end;
  suppose
A3: Y <> {};
    now
      let y;
      reconsider x = f.y as Element of X by A3,FUNCT_2:5;
      thus f.y = F.(x,x) by A1
        .= F.(f.y,f.y);
    end;
    hence thesis by A3,Th38;
  end;
end;
