Having had a quick look, does the following work? Put $x= \sum\_i x\_i/3$$x= \sum_i x_i/3$ and put
$$ y(t) = \sum\nolimits\_{i \in {R^1}_t} x_i = \sum\_i \eta\_i(t)x\_i $$
and $$ y(t) = \sum_{i \in R^1_t} x_i = \sum_i \eta_i(t)x_i $$ and try to substitute these into (3.2).
Observe that
$$ \begin{aligned} |x| + |y(t)| = | \frac13 \sum\_i x\_i | + | \sum\_i \eta\_i x\_i | & \leq | \frac13 \sum\_i x\_i | + | \sum\_i x\_i / 3 | + | \sum\_i (\eta\_i - 1/3)x\_i | \\\\ &\leq | \sum\_i x\_i | + | \sum\_i (\eta\_i - 1/3)x\_i | \end{aligned} $$
and $$ \begin{aligned} |x| + |y(t)| = | \frac13 \sum_i x_i | + | \sum_i \eta_i x_i | & \leq | \frac13 \sum_i x_i | + | \sum_i x_i / 3 | + | \sum_i (\eta_i - 1/3)x_i | \\ &\leq | \sum_i x_i | + | \sum_i (\eta_i - 1/3)x_i | \end{aligned} $$ and this should give what we want on the RHS of the formula you're asking about.
