A question about a detail in Bell's "Primer of Infinitesimal Analysis"

On p.35,36 of J.L. Bell's A Primer of Infinitesimal Analysis (2nd ed.), Bell uses the book's basic methods to derive the formula for the area of a circle based on the circumference. Where $s(x)$ is a function for the length of a certain portion of the circumference of a circle, and $C(x)$ is a formula for a portion of the area of a circle, and $\varepsilon$ is a nilpotent infinitesimal quantity, he derives the formula $$C'(x)=\frac{1}{2}rs'(x)$$ and goes on to say, "Since $C(0)=s(0)=0$, the Constancy Principle now yields $$C(x)=\frac{1}{2}rs(x)."$$

The "Constancy Principle" in question is that if $f$ is a real valued function with $f'(x)$ constantly $0$, then $f$ is constant; alternatively, $f$ is constant if $f(x+\varepsilon)=f(x)$ for all $x$ and infinitesimal $\varepsilon$.

My question is, how does $C(0)=s(0)=0$ allow use of the constancy principle, and why does this follow from it? The steps before and after this make total sense to me, but I don't see what's going on here, nor in similar steps in later proofs.

Answer

Let $f(x)=C(x)-\frac{1}{2}rs(x)$. Then $f'(x)=C'(x)-\frac{1}{2}rs'(x)=0$ for all $x$. By the Constancy Principle, $f$ is constant, so $f(x)=f(0)=C(0)-\frac{1}{2}rs(0)=0$ for all $x$. That is, $C(x)=\frac{1}{2}rs(x)$ for all $x$.