Let $$X$$ be a set with $$n \in \mathbb{N}$$ elements and let $$f: X \rightarrow \mathbb{N}$$ be a function from $$X$$ to the real numbers. Then we define the finite sum $$\sum_{x \in X} f(x)$$ as follows.
Select any [...] $$g$$ from $$\{i \in \mathbb{N} : 1 \le i \le n \}$$ to $$X$$. Such a [...] exists since $$X$$ is assumed to have $$n$$ elements. We then define:
$\sum_{x \in X} f(x) := \sum_{i=1}^{n}[...]$