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.

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