Lemma 2
Tag 0BXH, permanent. Cite it as [Bourbaki, Tag 0BXH].
General Topology, chapter VI, § 3. Real projective spaces
If there is a continuous mapping g of E onto A such that, for all x ∈ E, g(x) belongs to the equivalence class of x, then the quotient space A/R_A is homeomorphic to E/R.
This is Corollary 2 of Proposition 10 of Chapter I, § 3, no. 6.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.