Bourbaki

Remark

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Commutative Algebra, chapter VIII, § 7. Multiplicités

One can prove conversely (cf. p. 103, exerc. 4) that if A is regular and e(A/x)=δ1...δse(A/x) = \delta_1 ... \delta_s, then the sequence (ξ1,...,ξs)(\xi_1, ..., \xi_s) is completely secant.

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

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