  The braid group $B_n$ maps homomorphically into the Temperley-Lieb algebra
$\TL_n$. It was shown by Zinno that the homomorphic images of simple elements
arising from the dual presentation of the braid group $B_n$ form a basis for
the vector space underlying the Temperley-Lieb algebra $\TL_n$. In this paper,
we establish that there is a dual presentation of Temperley-Lieb algebras that
corresponds to the dual presentation of braid groups, and then give a simple
geometric proof for Zinno's theorem, using the interpretation of simple
elements as non-crossing partitions.
