  We consider a filtration $\mathbb{G}$ obtained as enlargement of a filtration
$\mathbb{F}$ by a filtration $\mathbb{H}$. We assume that all
$\mathbb{F}$-local martingales are represented by a martingale $M$ and all
$\mathbb{H}$-local martingales are represented by a martingale $N$. $M$ and $N$
are not necessarily quasi-left continuous processes and their jump times may
overlap. We first analyze the contribution of the accessible jump times of $M$
and $N$ to the Jacod's dimension of the space of the
$\mathcal{H}^1(\mathbb{G})$-martingales. Then we prove a new martingale
representation theorem on $\mathbb{G}$.
