reserve th, th1, th2, th3 for Real;

theorem
  sin(th1)*sin(th2)*sin(th3) = (1/4) *(sin(th1+th2-th3)+sin(th2+th3-th1)
  +sin(th3+th1-th2)-sin(th1+th2+th3))
proof
  sin(th1)*sin(th2)*sin(th3) =(-((1/2)*(cos(th1+th2)-cos(th1-th2))))*sin(
  th3) by Th29
    .=(1/2)*(cos(th1-th2)*sin(th3)-cos(th1+th2)*sin(th3))
    .=(1/2)*((1/2)*(sin((th1-th2)+th3)-sin((th1-th2)-th3)) -cos(th1+th2)*sin
  (th3)) by Th31
    .=(1/2)*((1/2)*(sin((th1-th2)+th3)-sin((th1-th2)-th3)) -(1/2)*(sin((th1+
  th2)+th3)-sin((th1+th2)-th3))) by Th31
    .=(1/(2*2))*((sin((th1-th2)+th3)+-sin((th1-th2)-th3)) +(sin((th1+th2)-
  th3)-sin((th1+th2)+th3)))
    .=(1/(2*2))*((sin((th1-th2)+th3)+sin(-((th1-th2)-th3))) +(sin((th1+th2)-
  th3)-sin((th1+th2)+th3))) by SIN_COS:31
    .=(1/(2*2))*(sin(th1+th2-th3)+sin(th2+th3-th1) +sin(th3+th1-th2)-sin(th1
  +th2+th3));
  hence thesis;
end;
