theorem
  F is commutative implies F.:(f,g) = F.:(g,f)
proof
  assume
A1: F is commutative;
  per cases;
  suppose
    Y = {};
    hence thesis;
  end;
  suppose
A2: Y <> {};
    now
      let y;
      reconsider x1 = f.y, x2 = g.y as Element of X by A2,FUNCT_2:5;
      thus (F.:(f,g)).y = F.(x1,x2) by A2,Th37
        .= F.(g.y,f.y) by A1;
    end;
    hence thesis by A2,Th38;
  end;
end;
