reserve th, th1, th2, th3 for Real;

theorem
  cos(th1+th2)*cos(th1-th2) = cos(th2)*cos(th2)-sin(th1)*sin(th1)
proof
  cos(th1+th2)*cos(th1-th2) = cos(th1)*cos(th1)-sin(th2)*sin(th2) by Th41
    .= (1-sin(th1)*sin(th1))-sin(th2)*sin(th2) by Th5
    .= (1-sin(th1)*sin(th1))-(1-cos(th2)*cos(th2)) by Th4
    .= cos(th2)*cos(th2)-sin(th1)*sin(th1);
  hence thesis;
end;
