On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all primes in $\mathcal{E}$. I quote:

By a simple inclusion-exclusion argument one expects that the number of such integers is about $$ x \prod_{p\in\mathcal{E}}\left(1-\frac{1}{p} \right). $$ This is provably always an upper bound, up to a constant: $$ S(T,T+x;\mathcal{E}) \ll x \prod_{p\in\mathcal{E}}\left(1-\frac{1}{p} \right), $$ and one gets the analogous lower bound $$ S(T,T+x;\mathcal{E}) \gg x \prod_{p\in\mathcal{E}}\left(1-\frac{1}{p} \right), $$ if $\mathcal{E}$ is a subset of the primes up to $x^{1/2-o(1)}$.

My question is simply:

QUESTION: Does anyone have references to proofs of the above bounds?

It seems these results are fairly standard knowledge, and the authors refer to the book "Opera de Cribro" by Friedlander and Iwaniec. However, they do not state any page numbers, so it is a bit hard to know exactly where to look. If the proofs are simple, perhaps someone would be able to jot them down here?

ADDED: While the provable lower bound is not as good as the provable upper bound, it seems reasonable they should be the same. This can be easily illustrated by plotting $$ \inf S(T,T+x, p_k\#) - x \cdot \prod_{p \in \mathcal P_k} \left( 1-\frac{1}{p} \right) \\ $$ and $$ \sup S(T,T+x, p_k\#) - x \cdot \prod_{p \in \mathcal P_k} \left( 1-\frac{1}{p} \right), $$ where $\mathcal P_k$ is the set of the $k$ first primes and $p_k\#=p_1 p_2 \dots p_k$ is the primorial. In the figure below, plots for both $k=4$ and $k=5$ are shown, and $x\in [0,p_k\#]$. Note that since $S(T+x, p_k\#)$ is periodic in $x$ with period $p_k\#$, we obtain the anti-symmetric property that $$ \inf S(T,T+x, p_k\#) - x \cdot \prod_{p \in \mathcal P_k} \left( 1-\frac{1}{p} \right)\\ = - \sup S(T, T+ p_k\#-x, p_k\#) + (p_k\#-x) \cdot \prod_{p \in \mathcal P_k} \left( 1-\frac{1}{p} \right), $$

This might be far fetched, but could this symmetric property be exploited in some way to better the lower bound?

Here is a different plot where $\mathcal{E}$ is the set of primes less or equal to $\sqrt{x}$, showing $$ \inf S(0,x,\mathcal{E}) - x \cdot \prod_{p \in \mathcal{E}} \left( 1-\frac{1}{p} \right) \\ $$ and $$ \sup S(0,x,\mathcal{E}) - x \cdot \prod_{p \in \mathcal{E}} \left( 1-\frac{1}{p} \right). $$