Let $FP(X)$ be the free paratopological group over a topological space $X$. For each nonnegative integer $n\in\mathbb{N}$, denote by $FP_n(X)$ the subset of $FP(X)$ consisting of all words of reduced length at most $n$, and $i_n$ by the natural mapping from $(X\oplus X^{-1}\oplus\{e\})^n$ to $FP_n(X)$. We prove that the natural mapping $i_2: (X\oplus X_d^{-1}\oplus\{e\})^2\to FP_2(X)$ is a closed mapping if and only if every neighborhood $U$ of the diagonal $\Delta_1$ in $X_d\times X$ is a member of the finest quasi-uniformity on $X$, where $X$ is a $T_1$-space and $X_d$ denotes $X$ when equipped with the discrete topology in place of its given topology.