reserve th, th1, th2, th3 for Real;

theorem
  sin(th1)*sin(th2)*cos(th3) = (1/4) *(-cos(th1+th2-th3)+cos(th2+th3-th1
  )+cos(th3+th1-th2)-cos(th1+th2+th3))
proof
  sin(th1)*sin(th2)*cos(th3) =(-((1/2)*(cos(th1+th2)-cos(th1-th2))))*cos(
  th3) by Th29
    .=(1/2)*(cos(th1-th2)*cos(th3)-cos(th1+th2)*cos(th3))
    .=(1/2)*((1/2)*(cos((th1-th2)+th3)+cos((th1-th2)-th3)) -cos(th1+th2)*cos
  (th3)) by Th32
    .=(1/2)*((1/2)*(cos((th1-th2)+th3)+cos((th1-th2)-th3)) -(1/2)*(cos((th1+
  th2)+th3)+cos((th1+th2)-th3))) by Th32
    .=(1/(2*2))*((-cos((th1+th2)-th3)+cos(-((th2-th1)+th3))) +(cos((th3+th1)
  +-th2)+-cos((th1+th2)+th3)))
    .=(1/(2*2))*((-cos((th1+th2)-th3)+cos((th2-th1)+th3)) +(cos((th3+th1)+-
  th2)+-cos((th1+th2)+th3))) by SIN_COS:31
    .=(1/(2*2))*(-cos(th1+th2-th3)+cos(th2+th3-th1) +cos(th3+th1-th2)-cos(
  th1+th2+th3));
  hence thesis;
end;
