reserve th, th1, th2, th3 for Real;

theorem
  sin(th1)*cos(th2)*cos(th3) = (1/4) *(sin(th1+th2-th3)-sin(th2+th3-th1)
  +sin(th3+th1-th2)+sin(th1+th2+th3))
proof
  sin(th1)*cos(th2)*cos(th3) =((1/2)*(sin(th1+th2)+sin(th1-th2)))*cos(th3)
  by Th30
    .=(1/2)*(sin(th1+th2)*cos(th3)+sin(th1-th2)*cos(th3))
    .=(1/2)*(((1/2)*(sin((th1+th2)+th3)+sin((th1+th2)-th3))) +sin(th1-th2)*
  cos(th3)) by Th30
    .=(1/2)*(((1/2)*(sin((th1+th2)+th3)+sin((th1+th2)-th3))) +((1/2)*(sin((
  th1-th2)+th3)+sin((th1-th2)-th3)))) by Th30
    .=(1/(2*2))*((sin((th1+th2)+th3)+sin((th1+th2)-th3)) +(sin((th1+-th2)+
  th3)+sin(-((th2-th1)+th3))))
    .=(1/(2*2))*((sin((th1+th2)+th3)+sin((th1+th2)-th3)) +(-sin((th2-th1)+
  th3)+sin((th3+th1)+-th2))) 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;
