reserve th, th1, th2, th3 for Real;

theorem
  sin(th1+th2)*sin(th1-th2)= cos(th2)*cos(th2)-cos(th1)*cos(th1)
proof
  sin(th1+th2)*sin(th1-th2) = (sin(th1)*sin(th1)-sin(th2)*sin(th2)) by Th37
    .= ((1-cos(th1)*cos(th1))-sin(th2)*sin(th2)) by Th4
    .= ((1---cos(th1)*cos(th1))---(1-cos(th2)*cos(th2))) by Th4
    .= cos(th2)*cos(th2)-cos(th1)*cos(th1);
  hence thesis;
end;
