reserve th, th1, th2, th3 for Real;

theorem Th18:
  cos(th1)-cos(th2)= -2*(sin((th1+th2)/2)*sin((th1-th2)/2))
proof
  cos(th1)-cos(th2)= cos(th1/2+th1/2)-cos(th2/2+th2/2)
    .= cos(th1/2)*cos(th1/2)-sin(th1/2)*sin(th1/2)-cos(th2/2+th2/2) by
SIN_COS:75
    .= ((cos(th1/2)*cos(th1/2)+(-sin(th1/2)*sin(th1/2)))+(1+(-1))) -(cos(th2
  /2)*cos(th2/2)-sin(th2/2)*sin(th2/2)) by SIN_COS:75
    .= ((cos(th1/2)*cos(th1/2)+(-sin(th1/2)*sin(th1/2)))+1) +((-1)+(-cos(th2
  /2)*cos(th2/2)+sin(th2/2)*sin(th2/2)))
    .= ((cos(th1/2)*cos(th1/2)+(-sin(th1/2)*sin(th1/2))) +(sin(th1/2)*sin(
  th1/2)+cos(th1/2)*cos(th1/2))) +((-1)+(-cos(th2/2)*cos(th2/2)+sin(th2/2)*sin(
  th2/2))) by SIN_COS:29
    .= (cos(th1/2)*cos(th1/2)+((sin(th1/2)*sin(th1/2) --(-sin(th1/2)*sin(th1
/2)))+cos(th1/2)*cos(th1/2))) +((-(sin(th2/2)*sin(th2/2)+cos(th2/2)*cos(th2/2))
  ) +(-cos(th2/2)*cos(th2/2)+sin(th2/2)*sin(th2/2))) by SIN_COS:29
    .= 2*((cos(th1/2)*cos(th1/2)*(1))-cos(th2/2)*cos(th2/2))
    .= 2*((cos(th1/2)*cos(th1/2)*(sin(th2/2)*sin(th2/2)+cos(th2/2)*cos(th2/2
  ))) -cos(th2/2)*cos(th2/2)) by SIN_COS:29
    .= 2*(cos(th1/2)*cos(th1/2)*(sin(th2/2)*sin(th2/2)) +(cos(th2/2)*cos(th2
  /2)*(cos(th1/2)*cos(th1/2)---1)))
    .= 2*(cos(th1/2)*cos(th1/2)*(sin(th2/2)*sin(th2/2)) +(cos(th2/2)*cos(th2
/2)*(cos(th1/2)*cos(th1/2) ---(cos(th1/2)*cos(th1/2)+sin(th1/2)*sin(th1/2)))))
  by SIN_COS:29
    .= -2*((sin(th1/2)*cos(th2/2)--(-cos(th1/2)*sin(th2/2))) *(sin(th1/2)*
  cos(th2/2)+cos(th1/2)*sin(th2/2)))
    .= -2*(sin(th1/2+th2/2)*(sin(th1/2)*cos(th2/2)-cos(th1/2)*sin(th2/2)))
  by SIN_COS:75
    .= -2*(sin((th1+th2)/2)*sin(th1/2-th2/2)) by SIN_COS:82
    .= -2*(sin((th1+th2)/2)*sin((th1-th2)/2));
  hence thesis;
end;
