An extension of a finitely presented group by a finitely presented group is finitely presented

Let $N$ be a normal subgroup of a group $G$ such that both $N$ and the quotient $G/N$ are finitely presented. Then $G$ is finitely presented.

A presentation of $G$ is obtained from generators of $N$ together with lifts of generators of $G/N$, with relations: the relations of $N$, the relations of $G/N$ read as words that land in $N$ and then expressed in the generators of $N$, and the conjugation relations expressing $x n x^{-1}$ in the generators of $N$ for each generator $x$ of the lift and each generator $n$ of $N$.

The weaker-looking hypothesis that $N$ is merely finitely generated and $G/N$ finitely presented does not suffice: taking $N = G$ would make every finitely generated group finitely presented.