# Two elementary theorems

I proved the following two elementary but useful theorems:

Theorem For every funcoids $f$, $g$:

1. If $\mathrm{im}\, f \supseteq \mathrm{im}\, g$ then $\mathrm{im}\, (g\circ f) = \mathrm{im}\, g$.
2. If $\mathrm{im}\, f \subseteq \mathrm{im}\, g$ then $\mathrm{dom}\, (g\circ f) = \mathrm{dom}\, g$.

Theorem For every reloids $f$, $g$:

1. If $\mathrm{im}\, f \supseteq \mathrm{im}\, g$ then $\mathrm{im}\, (g\circ f) = \mathrm{im}\, g$.
2. If $\mathrm{im}\, f \subseteq \mathrm{im}\, g$ then $\mathrm{dom}\, (g\circ f) = \mathrm{dom}\, g$.

See this Web page and especially this online article.