The Lefschetz number of the identity is Euler characteristic

Its rational cellular chain group $C_n=H_n(X^n,X^{n-1};\mathbb{Q})$ has dimension $c_n$ by [F4]. There are finitely many nonzero groups, since $X$ has finitely many cells. The identity map preserves every skeleton and induces the identity map of each relative group. Its cellular chain trace in degree $n$ is therefore the sum of the $c_n$ diagonal entries equal to one, namely $c_n$, with trace zero if $c_n=0$. By [F5], the induced homology map is the singular homology identity.

Apply [F3] to the identity chain map in step 1.1. Its homology trace sum is $L({\mathrm{i}\mathrm{d}}_X)$ by [F1], and its chain trace sum is ${\sum }_n(-1{)}^n{c}_n=\chi (X)$ by [F2]. This proves the equality over $\mathbb{Q}$ directly; no change-of-coefficients identification with integral ranks is needed.

If $X$ is nonempty and contractible, [F6] gives its rational homology as that of a point. For completeness, the singular chain group of a point is $\mathbb{Q}$ in each nonnegative degree, with its unique simplex as basis. In positive degree $n$ the boundary is multiplication by ${\sum }_{j=0}^n(-1{)}^j$, which is $1$ for even $n$ and $0$ for odd $n$. Thus in every positive degree the kernel equals the image from the next degree; in degree zero the boundary from degree one is zero. Consequently only $H_0=\mathbb{Q}$ survives. The identity therefore has trace $1$ in degree zero and no other nonzero traces, so [F1] gives $L=1$, and step 2.1 gives $\chi =1$. Every $x\in X$ satisfies ${\mathrm{i}\mathrm{d}}_X(x)=x$; nonemptiness supplies a fixed point without any choice family. This is consistent with the nonzero-Lefschetz sufficient condition, but the identity's fixed point is already explicit.

If $X$ is empty, there are no cells or singular simplices, so both sums are zero. If $X$ is a singleton, step 3.1 gives $1$; for a finite discrete space the same degree-zero identity matrix has one diagonal $1$ per point. Missing dimensions and zero cellular groups are covered by step 1.1. Degenerate singular simplices at a point are exactly the generators used in step 3.1 and do not produce unwanted higher homology. The proof is a finite alternating sum, with no infinite endpoint or limiting convention; the boundedness required by [F3] was verified in step 1.1. All basis choices in that theorem are finite, so this example is choice-free.