A new easy theorem in my draft.

Theorem $latex \mathrm{DOM} (g \circ f) \supseteq \mathrm{DOM} f$, $latex \mathrm{IM} (g \circ f) \supseteq \mathrm{IM} g$, $latex \mathrm{Dom} (g \circ f) \supseteq \mathrm{Dom} f$, $latex \mathrm{Im} (g \circ f) \supseteq \mathrm{Im} g$ for every composable morphisms $latex f$, $latex g$ of a category with restricted identities.

Proof $latex \mathcal{E}_{\mathcal{C}}^{Y, \mathrm{Dst} f} \circ \mathcal{E}_{\mathcal{C}}^{\mathrm{Dst} f, Y} \circ g \circ f = g \circ f \Leftarrow \mathcal{E}_{\mathcal{C}}^{Y, \mathrm{Dst} f} \circ \mathcal{E}_{\mathcal{C}}^{\mathrm{Dst} f, Y} \circ g = g$ and it implies $latex \mathrm{IM} (g \circ f) \supseteq \mathrm{IM} g$. The rest follows easily.

Corollary $latex \mathrm{dom} (g \circ f) \sqsubseteq \mathrm{dom} f$, $latex \mathrm{im} (g \circ f) \sqsubseteq \mathrm{im} g$ whenever $latex \mathrm{dom}$/$latex \mathrm{im}$ are defined.

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