Projection is Epimorphism

Theorem
Let $$\left({S, \circ}\right)$$ be the external direct product of the algebraic structures $$\left({S_1, \circ_1}\right)$$ and $$\left({S_2, \circ_2}\right)$$.


 * $$pr_1$$ is an epimorphism from $$\left({S, \circ}\right)$$ to $$\left({S_1, \circ_1}\right)$$;
 * $$pr_2$$ is an epimorphism from $$\left({S, \circ}\right)$$ to $$\left({S_2, \circ_2}\right)$$.

where $$pr_1$$ and $$pr_2$$ are the first and second projection of $$\left({S, \circ}\right)$$.

Generalized Version
Let $$\left({S, \circ}\right)$$ be the external direct product of the algebraic structures $$\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_k, \circ_k}\right), \ldots, \left({S_m, \circ_m}\right)$$.

Then for each $$j \in \left[{1 \,. \, . \, n}\right]$$, $$pr_j$$ is an epimorphism from $$\left({S, \circ}\right)$$ to $$\left({S_j, \circ_j}\right)$$

where $$pr_j: \left({S, \circ}\right) \to \left({S_j, \circ_j}\right)$$ is the $j$th projection from $$\left({S, \circ}\right)$$ to $$\left({S_j, \circ_j}\right)$$.