Let $p_1, q_1, p_2, q_2, \, \ldots, \, p_N, q_N \in (0, 1]$ each be a D993: Real number such that

(i)	\begin{equation} \sum_{n = 1}^N p_n = 1 \end{equation}
(ii)	\begin{equation} \sum_{n = 1}^N q_n = 1 \end{equation}

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.

Then

(1)	\begin{equation} - \sum_{n = 1}^N p_n \log_a p_n \leq - \sum_{n = 1}^N p_n \log_a q_n \end{equation}
(2)	\begin{equation} - \sum_{n = 1}^N p_n \log_a p_n = - \sum_{n = 1}^N p_n \log_a q_n \quad \iff \quad \forall \, n \in \{ 1, 2, \, \ldots, \, N \} : p_n = q_n \end{equation}