reserve th, th1, th2, th3 for Real;

theorem
  cos(th1+th2)*sin(th1-th2) = sin(th1)*cos(th1)-sin(th2)*cos(th2)
proof
  cos(th1+th2)*sin(th1-th2) = sin(th1+(-th2))*cos(th1-(-th2))
    .= sin(th1)*cos(th1)+sin(-th2)*cos(-th2) by Th39
    .= sin(th1)*cos(th1)+sin(-th2)*cos(th2) by SIN_COS:31
    .= sin(th1)*cos(th1)+(-sin(th2))*cos(th2) by SIN_COS:31
    .= sin(th1)*cos(th1)-(sin(th2)*cos(th2));
  hence thesis;
end;
