Definition 4

Tag 056Q, permanent. Cite it as [Bourbaki, Tag 056Q].

Algebra, chapter I, § 2. Identity element; cancellable elements; invertible elements

Given a law of composition $\top$ on a set E, the mapping $x \mapsto a \top x$ (resp. $x \mapsto x \top a$ ) of E into itself is called left translation (resp. right translation) by an element $a \in E$ .

On passing to the opposite law, left translations become right translations and conversely.

Let $\gamma_a, \delta_a$ (or $\gamma(a), \delta(a)$ ) denote the left and right translations by $a \in E$ ; then

\gamma_a(x) = a \top x, \qquad \delta_a(x) = x \top a.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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