Because $N = p_1 p_2 \cdots p_n + 1 \ge 3 > 1$, there are two cases for $N$, just as Euclid distinguishes in Book IX, Proposition 20: "Then $EF$ is either prime or not." If $N$ is prime, then $q = N$ is a prime divisor of $N$ (since $N \mid N$). If $N$ is composite, Euclid invokes Book VII, Proposition 31: "Any composite number is measured by some prime number."

Why does Book VII, Proposition 31 hold? Euclid proves it by infinite descent: if $N$ is composite, it has a divisor $d_1$ with $1 < d_1 < N$. If $d_1$ is prime, we are done; if $d_1$ is composite, it has a strictly smaller divisor $1 < d_2 < d_1 < N$, and so on. As Euclid writes, "if it is not found, then an infinite sequence of numbers measures the number $A$, each of which is less than the other, which is impossible in numbers." Equivalently, in modern terms, the smallest divisor $q > 1$ of $N$ must be prime (otherwise a smaller divisor of $q$ would also divide $N$). Either way, $N$ has at least one prime factor $q$.