As stated in the comments, you get the symbols in mathmode simply by writing them down. Packages like amsmath and amssymb support you.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\newtheorem{theorem}{THEOREM}
\newtheorem{proof}{PROOF}
\begin{document}
\begin{theorem}
If an operator has both Left Identity and
Right Identity then it is \emph{UNIQUE}.
\end{theorem}
\begin{proof}
Let $e_{l}$ is left identity 
$\therefore e_{l} * e_{r} \implies e_{r}$
\end{proof}
\end{document}

If you add \usepackage{amssymb} to your preamble (or any package which loads this), you can just write \therefore in maths mode. As Willie Wong says, if you have even a minimally functional installation of LaTeX, you already have this package.

If you are using non-default text fonts, you might want to use a different set of fonts for maths, too, but \therefore will be available in any reasonably comprehensive configuration.

Insert a {} between \therefore and the & alignment symbol. This restores the proper space of a binary relation.

\begin{align*}
\text{Characteristic equation: } &r^2 + \lambda^2 c^2 = 0\\
\therefore{}&r = \pm\lambda ci
\end{align*}   

produces

instead of

This approach is useful whenever we use symbols defined with \mathrel or \mathbin followed by a & alignment symbol. In this case \therefore is defined in amssymb.sty with \mathrel.

There are multiple symbols commonly used to represent "such that"

1. $\backepsilon$ \backepsilon or $\ni$ \ni
2. $\mid$ \mid
3. $:$ :

For "therefore", I usually see $\therefore$ \therefore