Projection is Surjection

Theorem
For two non-empty sets $$S$$ and $$T$$, the first projection and second projection are both surjections.

Generalized Version
For all non-empty sets $$S_1, S_2, \ldots, S_j, \ldots, S_n$$, the $j$th projection $$\operatorname{pr}_j$$ on $$\prod_{i=1}^n S_i$$ is a surjection.

Proof
Consider the $$j$$th projection.

As long as none of $$S_1, S_2, \ldots, S_n$$ is the empty set, then:


 * $$\forall x \in S_j: \exists \left({s_1, s_2, \ldots, s_{j-1}, x, s_{j+1}, \ldots, s_n}\right) \in \prod_{k=1}^n S_k: \operatorname{pr}_j \left({\left({s_1, s_2, \ldots, s_{j-1}, x, s_{j+1}, \ldots, s_n}\right)}\right) = x$$

and that's all we need to show.