On the paving size of a subfactor

Given an inclusion of II${}_1$ factors $N\subset M$ with finite Jones index, $[M:N]<\mathrm{\infty }$, we prove that for any $F\subset M$ finite and $\epsilon >0$, there exists a partition of $1$ with $r\le \lceil 16{\epsilon }^{-2}\rceil$ $\cdot \lceil 4[M:N]{\epsilon }^{-2}\rceil$ projections $p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_r\in N$ such that $\mathrm{\parallel }{\sum }_{i=1}^r{p}_{i}x{p}_i-E_{N^{\mathrm{\prime }}\cap M}(x)\mathrm{\parallel }\le \epsilon \mathrm{\parallel }x-E_{N^{\mathrm{\prime }}\cap M}(x)\mathrm{\parallel }$, $\mathrm{\forall }x\in F$ (where $\lceil \beta \rceil$ denotes the least integer $\ge \beta$). We consider a series of related invariants for $N\subset M$, generically called {\it paving size}.