Another proof that the two color bipartite Ramsey number is $O(2^t)$

For positive integers $t$ and $q$ let $b_q(t)$ be the smallest integer $n$ so that any coloring of the edges of the complete bipartite graph $K_{n,n}$ with $q$ colors yields a monochromatic copy of $K_{t,t}$. We give an independent proof that $b_2(t)\le 128\cdot 2^t$ for every positive integer $t$.