Corollary 3
Tag 05DH, permanent. Cite it as [Bourbaki, Tag 05DH].
Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups
Let G be a nilpotent group and H a subgroup of G. If G = H.(G, G), then G = H.
Every subgroup N of G which contains H and such that G/N is commutative contains H.(G, G). Corollary 3 thus follows from Corollary 2.
Corollary 3 can also be formulated thus: let X be a subset of G. For X to generate G, it is necessary and sufficient that the image of X in G/D(G) generate G/D(G).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.