reserve th, th1, th2, th3 for Real;

theorem Th41:
  cos(th1+th2)*cos(th1-th2)= cos(th1)*cos(th1)-sin(th2)*sin(th2)
proof
  cos(th1+th2)*cos(th1-th2) = (cos(th1)*cos(th2)-sin(th1)*sin(th2))*cos(
  th1-th2) by SIN_COS:75
    .= (cos(th1)*cos(th2)+-sin(th1)*sin(th2)) *(cos(th1)*cos(th2)+sin(th1)*
  sin(th2)) by SIN_COS:83
    .= (cos(th1)*cos(th1)*(cos(th2)*cos(th2)) +cos(th1)*cos(th2)*(sin(th1)*
sin(th2)) +-((sin(th1)*sin(th2))*(cos(th1)*cos(th2))) +-(((sin(th1)*sin(th1))*
  sin(th2))*sin(th2)))
    .= (cos(th1)*cos(th1)*(1---sin(th2)*sin(th2)) +-((sin(th1)*sin(th1))*(
  sin(th2)*sin(th2)))) by Th5
    .= (cos(th1)*cos(th1)*(1) +((sin(th2)*sin(th2))*(-((cos(th1)*cos(th1)+
  sin(th1)*sin(th1))))))
    .= (cos(th1)*cos(th1)*(1)+(sin(th2)*sin(th2)*(-1))) by SIN_COS:29
    .= cos(th1)*cos(th1)-sin(th2)*sin(th2);
  hence thesis;
end;
