Block bases of the Haar system as complemented subspaces of $L_p$, $2<p<\infty$

It is shown that the span of $\{a_ih_i\oplus b_i e_i \}^n_{i=1}$, where $\{h_i\}$ is the Haar system in $L_p$and $\{e_i\}$ the canonical basis of $\ell_p$, is well isomorphic to a well complemented subspace of $L_p, 2<p<\infty$. As a consequence we get that there is a rearrangement of the (initial segments of the) Haar system in $L_p, 2<p<\infty$, any block basis of which is well isomorphic to a well complemented subspace of $L_p$.