What shown to follow is a theorem taken from the text Analysis on Manifolds written by James Munkres.
So at the page 11 of this document, if you like you can read the proof of the mentioned theorem.
Now more generally I ask if this result holds.
Conjecture
Let $A$ be open in $\Bbb R^{n+k}$ and let be $f:A\rightarrow\Bbb R^n$ be of class $C^r$. Let $M$ be the set of all $x$ such that $$ f_1(x)=0,...,f_{n-(h+1)}(x)=0,f_{n-h}(x)>0,...,f_{n-i}(x)=0,....,f_n(x)>0 $$ for any $i=1,...,h$ and assume that $M$ is not empty and that $Df$ has rank $(n-h)$ on $M$. Then $M$ is a $k+h$ manifold without boundary in $\Bbb R^{n+k}$. Furthermore if $N$ is the set of all $x$ for which $$ f_1(x)=0,...,f_{n-(h+1)}(x)=0,f_{n-h}(x)\ge 0,...,f_n(x)\ge 0 $$ and if the $Df$ has rank $n+(h-1)$ at each poin of $N$, then $N$ is a $k+(h+1)$ manifold and $\partial N=M$.
So substantially I ask when the solutions set of the constraint system $$ \begin{cases}f_1(x)=0\\ \vdots\\ f_{n-(h+1)}(x)=0\\ f_{n-h}(x)\ge 0\\ \vdots\\ f_n(x)\ge 0\end{cases} $$ is a $k+(h+1)$ manifold.
So is my conjecture true? And if it is true how prove it?