\begin{definition}[Definition:Cartesian Product/Coordinate]
Let $\ds \prod_{i \mathop \in I} S_i$ be a cartesian product.
Let $j \in I$, and let $s = \sequence {s_i}_{i \mathop \in I} \in \ds \prod_{i \mathop \in I} S_i$.
Then $s_j$ is called the '''$j$th coordinate of $s$'''.
If the indexing set $I$ consists of ordinary numbers $1, 2, \ldots, n$, one speaks about, for example, the '''first''', '''second''', or '''$n$th coordinate'''.
For an element $\tuple {s, t} \in S \times T$ of a binary cartesian product, $s$ is the '''first coordinate''', and $t$ is the '''second coordinate'''.
\end{definition}