Proposition 6
Tag 0BBS, permanent. Cite it as [Bourbaki, Tag 0BBS].
General Topology, chapter I, § 11. Connectedness
Every quotient space of a connected space is connected.
This is an immediate consequence of Proposition 4 of no. 2.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.