reserve th, th1, th2, th3 for Real;

theorem Th31:
  cos(th1)*sin(th2)= (1/2)*(sin(th1+th2)-sin(th1-th2))
proof
  cos(th1)*sin(th2) = (1/2)*(sin(th1)*cos(th2)+cos(th1)*sin(th2) -(sin(th1
  )*cos(th2)-cos(th1)*sin(th2)))
    .= (1/2)*(sin(th1+th2)-(sin(th1)*cos(th2)+cos(th1)*(-sin(th2)))) by
SIN_COS:75
    .= (1/2)*(sin(th1+th2)-(sin(th1)*cos(th2)+cos(th1)*sin(-th2))) by
SIN_COS:31
    .= (1/2)*(sin(th1+th2)-(sin(th1)*cos(-th2)+cos(th1)*sin(-th2))) by
SIN_COS:31
    .= (1/2)*(sin(th1+th2)-sin(th1+(-th2))) by SIN_COS:75
    .= (1/2)*(sin(th1+th2)-sin(th1-th2));
  hence thesis;
end;
