$\textbf{Background.}$ I'm trying to apply Lojasiewicz's inequality to a specific function, $f$, and the Euclidean distance function $d$. Hence, I need to prove that $f$ and $d$ are subanalytic. Moreover, let $Q = [0,1]^2$. Specifically, $f:Q \to \mathbf{R}$ is real-analytic and non-negative. Moreover, $f$ is bounded on $Q$.
$\textbf{Definitions:}$
*Let $\mathcal{A}$ be a ring of real-valued functions defined on a set $E$. Let $S(\mathcal{A})$ denote the subsets of $E$ which are "described by" $\mathcal{A}$; i.e., the smallest family of subsets of $E$ containing all $\{x: f(x) > 0 \}$, $f\in \mathcal{A}$, which is stable under finite intersection, finite union, and complement.
Equivalently, $S(\mathcal{A})$ means the subsets of $E$ of the form $$ X = \bigcup_{i=1}^{p}\bigcap_{j=1}^{q}X_{ij}, $$ where each $X_{ij}$ is either $\{x: f_{ij}(x) > 0\}$ or $\{x: f_{ij}(x)=0\}$, $f_{ij}\in \mathcal{A}$. We say that $X$ is "described by" $\{f_{ij}\}$.
*Let $M$ denote a real analytic manifold. If $U$ is an open subset of $M$, let $\mathcal{O}(U)$ denote the ring of real analytic functions on $U$.
*A subset $X\subseteq M$ is semianalytic if for each $m\in M$ there exists a neighborhood $U_m$ such that $X\cap U_m \in S(\mathcal{O}(U_m))$.
*$X\subseteq M$ is subanalytic if for each point $m\in M$ there exists a neighborhood $U_m$ such that $X \cap U_m$ is a projection of a relatively compact semianalytic set. (i.e. there is a real analytic manifold $N$ and relatively compact semianalytic set $A\subseteq M\times N$ such that $X\cap U_m = \pi(A)$, where $\pi: M\times N \to M$ is the projection map.
*Let $X\subseteq M$ and $N$ be real analytic manifolds. A mapping $f: X \to N$ is subanalytic if its graph is subanalytic in $M\times N$.
$\textbf{Question.}$ Is $f$ subanalytic on compact subsets $K \subseteq Q$?
$\textbf{Ideas.}$ Let $$ \Gamma(f) = \{(x,f(x)): x\in K\} $$ denote the graph of $f$. We must show that for each $m\in Q\times \mathbf{R}$ there exists a neighborhood $U_m$ such that $\Gamma(f)\cap U_m$ is a projection of a relatively compact semianalytic set. In other words, we must find a real manifold $L$ and relatively compact semianalytic set $A$ such that $A \subseteq Q\times \mathbf{R}\times L$ where, $$ \Gamma(f)\cap U_m = \pi(A), $$ and $\pi:Q\times \mathbf{R}\times L \to Q\times \mathbf{R}$.
Therefore, let $L = \mathbf{R}$. It is a real analytic manifold. Additionally, define $A = \{(x,f(x), y): x\in Q, y\in (0,1)\}$. We must show that $A$ is relatively compact and semianalytic. $A$ is relatively compact as it is bounded and its closure is compact as $f$ is bounded. Hence, we must show that $A$ is semianalytic. In other words, for each point $n \in Q\times \mathbf{R}\times L$ we must find a neighborhood $N_n$ such that $A \cap N_n \in S(\mathcal{O}(N_n))$.
However, at this point I'm kind of fumbling. Unfortunately, this is my first exposure to real-analytic geometry.
Then to finish off the proof you just define the projection map $\pi: Q\times \mathbf{R} \times L \to Q\times \mathbf{R}$ as $\pi(x,y,z) = (x,y)$. Hence, $\pi(A) = \{(x,f(x)): x\in K\}$.
However, I'm a bit stuck showing that $A$ is semianalytic. Somehow I need to show that it is "described by" real analytic functions. Any help would be appreciated.