  We show that if an Ahlfors-David regular set $E$ of dimension $k$ has Big
Pieces of Big Pieces of Lipschitz Graphs (denoted usually by $BP(BP(LG))$),
then $E\subset \tilde{E}$ where $\tilde{E}$ is Ahlfors-David regular of
dimension $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by
$BP(LG)$. Our results are quantitative and, in fact, are proven in the setting
of a metric space for any family of Ahlfors-David regular sets $\mathcal{F}$
replacing $LG$. A simple corollary is the stability of the BP operator after 2
iterations. This was previously only known in the Euclidean setting for the
case $\mathcal{F}= LG$ with substantially more complicated proofs.
