Proposition 7
Tag 0563, permanent. Cite it as [Bourbaki, Tag 0563].
Algebra, chapter I, § 1. Laws of composition; associativity; commutativity
Let E be a magma and R an equivalence relation on E compatible with the law on E. For an equivalence relation S on E/R to be compatible with the quotient law it is necessary and sufficient that S be of the form T/R where T is an equivalence relation on E implied by R and compatible with the law on E. The canonical mapping of E/T onto (E/R)/(T/R) (Set Theory, II, § 6, no. 7) is then a magma isomorphism.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.