reserve th, th1, th2, th3 for Real;

theorem
  cos(th1)*cos(th2)*cos(th3) = (1/4) *(cos(th1+th2-th3)+cos(th2+th3-th1)
  +cos(th3+th1-th2)+cos(th1+th2+th3))
proof
  cos(th1)*cos(th2)*cos(th3) =((1/2)*(cos(th1+th2)+cos(th1-th2)))*cos(th3)
  by Th32
    .=(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((th1+th2)-th3)) +(cos((th3+th1)+-
  th2)+cos(-((th2-th1)+th3))))
    .=(1/(2*2))*((cos((th1+th2)+th3)+cos((th1+th2)-th3)) +(cos((th3+th1)-th2
  )+cos((th2+th3)+-th1))) 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;
