I think I found out.
$$\displaystyle (1-\frac{1}{p^{s+1}}-\frac{1}{p^{t+1}}+\frac{1}{p^{t+s+1}})^k=\sum_{m_1,...,m_4}\frac{k!}{m_1!,...m_4!}1^{m_1}p^{^-(s+1)m_2}p^{-(t+1)m_3}p^{-(s+t+1)m_4} $$
Right here we get $ \displaystyle (1-\frac{1}{p^{s+1}}-\frac{1}{p^{t+1}}+\frac{1}{p^{t+s+1}})$ and rest of it is $O(1/p^2)$.
Since
$$ \displaystyle \frac{(1-p^{-(s+1)})(1-p^{-(t+1)})}{(1-p^{-(2+s+t)})}=(1-p^{-(s+1)})(1-p^{-(t+1)})(1+p^{-(2+s+t)}+O(1/p^2)). $$
equation holds.