I am reading the following proof of integral inequality lemma:
Let $b(t)$ and $f(t)$ be continuous functions for $t \geq \alpha$, and $v(t)$ is a differentiable function for $t \geq \alpha$. Suppose that
\begin{equation}
v'(t) = b(t)v(t) + f(t), \quad t \geq \alpha ,\quad v(\alpha) \leq v_0. (1.5)
\end{equation}
Then \begin{equation}v(t) \leq v_0 e^{\int_{\alpha}^{t}b(s)ds} + \int_{\alpha}^{t}f(s)e^{\int_{s}^{t}b(\tau)d\tau}ds. \quad (1.6) \end{equation}
I understand the proof but there is a remark saying, the result of the following lemma remains valid if $\leq$ is replaced by $\geq$ in both (1.5) and (1.6).
My question as I do not really understand english is, when they say the result of lemma one remains valid what does he mean, by replacing the inequality signs should I only prove that $$v(t) \geq v_0 e^{\int_{\alpha}^{t}b(s)ds} + \int_{\alpha}^{t}f(s)e^{\int_{s}^{t}b(\tau)d\tau}ds $$ ?