reserve th, th1, th2, th3 for Real;

theorem Th17:
  cos(th1)+cos(th2)= 2*(cos((th1+th2)/2)*cos((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)-(sin(th2/2)*sin(th2/2)*(1)))
    .= 2*(cos(th1/2)*cos(th1/2)-(sin(th2/2)*sin(th2/2) *(cos(th1/2)*cos(th1/
  2)+sin(th1/2)*sin(th1/2)))) by SIN_COS:29
    .= 2*(cos(th1/2)*cos(th1/2)*(1---sin(th2/2)*sin(th2/2)) +-sin(th2/2)*sin
  (th2/2)*(sin(th1/2)*sin(th1/2)))
    .= 2*(cos(th1/2)*cos(th1/2)*((cos(th2/2)*cos(th2/2)+sin(th2/2)*sin(th2/2
  )) ---sin(th2/2)*sin(th2/2)) +-sin(th2/2)*sin(th2/2)*(sin(th1/2)*sin(th1/2)))
  by SIN_COS:29
    .= 2*((cos(th1/2)*cos(th2/2)+sin(th1/2)*sin(th2/2)) *(cos(th1/2)*cos(th2
  /2)+-sin(th1/2)*sin(th2/2)))
    .= 2*((cos(th1/2-th2/2)) *(cos(th1/2)*cos(th2/2)-sin(th1/2)*sin(th2/2)))
  by SIN_COS:83
    .= 2*((cos((th1-th2)/2)*(cos(th1/2+th2/2)))) by SIN_COS:75
    .= 2*(cos((th1+th2)/2)*cos((th1-th2)/2));
  hence thesis;
end;
