[proofplan]
Use the upper and lower Rosser-Iwaniec weights assumed in the statement to compare the sifted count with two weighted divisor sums. Substitute the decomposition $|A_d| = X\omega(d)/d + R_d$ into those sums. The assumed main-term evaluations give the contributions $XV(z)F(s)$ and $XV(z)f(s)$, while the coefficient bounds $|\lambda_d^\pm| \le 1$ control the total remainder by the stated level-$D$ remainder sum.
[/proofplan]
[step:Majorize the sifted count by the upper weighted sum]
For each term $a$ of the finite sequence $A$, the assumed upper Rosser-Iwaniec inequality gives
\begin{align*}
\mathbb{1}_{\gcd(a,P(z)) = 1}
\le
\sum_{\substack{d \le D,\ d \mid P(z),\ d \mid a}} \lambda_d^+.
\end{align*}
Summing this inequality over all terms $a$ of $A$, counted with multiplicity, and interchanging the finite sums gives
\begin{align*}
S(A,z)
\le
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ |A_d|.
\end{align*}
Using $|A_d| = X\omega(d)/d + R_d$, we obtain
\begin{align*}
S(A,z)
\le
X\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ \frac{\omega(d)}{d}
+
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ R_d.
\end{align*}
Since $|\lambda_d^+| \le 1$ for every $d$, the remainder term satisfies
\begin{align*}
\left|\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ R_d\right|
\le
\sum_{\substack{d \le D,\ d \mid P(z)}} |R_d|.
\end{align*}
The assumed upper main-term evaluation therefore yields
\begin{align*}
S(A,z)
\le
XV(z)\{F(s) + o(1)\}
+
O_C\left(\sum_{\substack{d \le D,\ d \mid P(z)}} |R_d|\right),
\end{align*}
with the same uniformity in $s$ as in the weight evaluation.
[guided]
The purpose of the upper weights is to replace the difficult condition $\gcd(a,P(z)) = 1$ by a divisor sum whose terms can be counted using the hypotheses on $A_d$. For each $a \in A$, the statement assumes
\begin{align*}
\mathbb{1}_{\gcd(a,P(z)) = 1}
\le
\sum_{\substack{d \le D,\ d \mid P(z),\ d \mid a}} \lambda_d^+.
\end{align*}
Summing over $a \in A$ gives
\begin{align*}
S(A,z)
=
\sum_{a \in A} \mathbb{1}_{\gcd(a,P(z)) = 1}
\le
\sum_{a \in A}\sum_{\substack{d \le D,\ d \mid P(z),\ d \mid a}} \lambda_d^+.
\end{align*}
The sums are finite, so we may interchange their order. For a fixed $d$, the number of elements $a \in A$ divisible by $d$ is precisely $|A_d|$. Hence
\begin{align*}
S(A,z)
\le
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ |A_d|.
\end{align*}
Now the hypothesis on the distribution of $A$ in residue classes gives
\begin{align*}
|A_d| = X\frac{\omega(d)}{d} + R_d.
\end{align*}
Substituting this identity separates the main term from the error:
\begin{align*}
S(A,z)
\le
X\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ \frac{\omega(d)}{d}
+
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ R_d.
\end{align*}
The coefficient condition $|\lambda_d^+| \le 1$ controls the second sum by
\begin{align*}
\left|\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ R_d\right|
\le
\sum_{\substack{d \le D,\ d \mid P(z)}} |R_d|.
\end{align*}
Finally, the assumed upper Rosser-Iwaniec main-term estimate gives
\begin{align*}
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^+ \frac{\omega(d)}{d}
\le
V(z)\{F(s) + o(1)\}.
\end{align*}
Combining the last two estimates proves the stated upper bound, with uniformity for $s$ in compact subintervals inherited from the assumed weight estimate.
[/guided]
[/step]
[step:Minorize the sifted count by the lower weighted sum]
For each term $a$ of the finite sequence $A$, the assumed lower Rosser-Iwaniec inequality gives
\begin{align*}
\mathbb{1}_{\gcd(a,P(z)) = 1}
\ge
\sum_{\substack{d \le D,\ d \mid P(z),\ d \mid a}} \lambda_d^-.
\end{align*}
Summing over $a \in A$, interchanging finite sums, and using $|A_d| = X\omega(d)/d + R_d$ gives
\begin{align*}
S(A,z)
\ge
X\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^- \frac{\omega(d)}{d}
+
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^- R_d.
\end{align*}
Since $|\lambda_d^-| \le 1$, we have
\begin{align*}
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^- R_d
\ge
-\sum_{\substack{d \le D,\ d \mid P(z)}} |R_d|.
\end{align*}
The assumed lower main-term evaluation gives
\begin{align*}
\sum_{\substack{d \le D,\ d \mid P(z)}} \lambda_d^- \frac{\omega(d)}{d}
\ge
V(z)\{f(s) + o(1)\}.
\end{align*}
Therefore
\begin{align*}
S(A,z)
\ge
XV(z)\{f(s) + o(1)\}
-
O_C\left(\sum_{\substack{d \le D,\ d \mid P(z)}} |R_d|\right),
\end{align*}
again uniformly for $s$ in the prescribed compact subintervals.
[/step]
[step:Identify when the lower main term is positive]
By the defining normalization of the lower linear sieve function, $f(s)=0$ for $0<s\le 2$, and by the standard positivity property included in the statement, $f(s)>0$ for $s>2$. Thus the lower bound can have a positive main term only in the range where the lower sieve function is positive, namely $s>2$. This is exactly the final assertion of the theorem.
[/step]
