$k $ is assumed to be the set of real numbers, $\mathbb{R}$ throughout the paper.

A formula of the form $$(Q_{1}x_{1}),. . .,(Q_{n}x_{n}) [\psi(y_{1},. . .,y_{m},x_{1},. . .,x_{n})]$$ is called a *prenex* formula, where $Q_{i} \in \{ \exists, \forall \}$ and $\psi$ is a quantifier-free formula in $x_1,,. . .,x_n,y_1,. . .,y_m$.

$V \subset \mathbb{R}$ is said to be a semialgebraic set if there exists $ f_1,. . .,f_s \in \mathbb{R} [x_1,. . .,x_n]$ and $g_1,. . .,g_t \in \mathbb{R} [x_1,. . .,x_n]$ such that $V = \{ (a_1,. . .,a_n) \in \mathbb{R} : f_i(a_1,. . .,a_n) = 0, i=1,. . . ,s, g_j(a_1,. . .,a_n) \geq 0, i=1,. . .,t\}$.

A connected subset of $\mathbb{R}^{n}$ is called a ‘region’ in $\mathbb{R}^{n}$. Consider a region $R$ and functions $f_i : \mathbb{R}^n \longrightarrow \mathbb{R}, i=1,. . .,l$ satisfying, $f_{1} < f_{2} < ... < f_{l}$. This ordering ensures that these functions do not intersect each other. Graph of $f_{i}$ is called ‘$f_{i}$-section’. In other words, ‘$f_{i}$-section is the set containing all points of the form $(a,b)$, where a $\in \mathbb{R}^{n-1}$ and $b = f(a)$. An ‘(f$_{i}$,f$_{i+1}$)-sector’ is the set of all points $(a,b)$, where a $\in \mathbb{R}^{n-1}$ and $f_{i}(a)$ $< b 