If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in \mathfrak{U}$. Let $\pi: X\rightarrow X/R$ denote the quotient map and put $(Y,\mathfrak{W})= (X/R,\mathfrak{U}/R)$.

$(Y,\mathfrak{W})$ is called he separated uniform space associated with $(X,\mathfrak{U})$.

I am trying to show that $(Y,\mathfrak{W})$ is a hausdorff uniform space.

$\mathfrak{L}=\{ (\pi\times\pi)(M):M\in \mathfrak{U}\}$ is a basis for a uniformity ($\mathfrak{W}$) on $X/R$.

how can we show that: for every $(\pi\times\pi)(M)\in \mathfrak{L}$; $(\pi\times\pi)(N)\circ(\pi\times\pi)(N) \subset (\pi\times\pi)(M)$ for some $(\pi\times\pi)(N) \in \mathfrak{L}$.

and why $(\pi\times\pi)^{-1}((\pi\times\pi)(M))= R\circ M\circ R$; for every $M\in \mathfrak{U}$ ?

$(Y,\mathfrak{W})$ Used for construct the hausdorff completion of a hausdorff uniform space $(X,\mathfrak{U})$.