Continuity of the Projections and of the Distance Function on a Product Metric Space

Let $\mathbb{R}$ be the set of real numbers with the addition, multiplication and order $\le$ of its ordered field structure, regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line. Then the following hold.

1. (Projections) Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $d_{X\times Y}$ be the product metric on $X\times Y$, a metric by claim 1 of The Product Metric is a Metric, and let $S\subseteq X\times Y$. Then the map $\pi_X:S\to X$ with $\pi_X((x,y))=x$ is continuous on $S$ relative to $S$, and the map $\pi_Y:S\to Y$ with $\pi_Y((x,y))=y$ is continuous on $S$ relative to $S$.

2. (Distance function) Let $(Z,d_Z)$ be a metric space, let $d_{Z\times Z}$ be the product metric on $Z\times Z$ obtained from $d_Z$ and $d_Z$, and let $T\subseteq Z\times Z$. Then the map $\rho:T\to\mathbb{R}$ with $\rho((x,y))=d_Z(x,y)$ is continuous on $T$ relative to $T$, as a map into the metric space $(\mathbb{R},d_{\mathbb{R}})$.