Having had a quick look, does the following work? Put $x= \sum_i x_i/3$ and put $$ y(t) = \sum_{i \in R^1_t} x_i = \sum_i \eta_i(t)x_i $$ 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 this should give what we want on the RHS of the formula you're asking about.
