Lemma 3

Let M be a semisimple A-module. Let B be the bicommutant of the A-module M. Then every A-submodule of M is a B-submodule of M.

Let N be an A-submodule of M. By Corollary 2 of VIII, p. 56, there exists a projector $p$ of the A-module M with image N; it belongs to $A'_M$. Since we have the relation $pb=bp$ for every $b\in B$, we obtain that N is a B-submodule of M.

Let us conclude the proof of Theorem 3. Suppose that $c$ belongs to $A''_M$, and let F = $\{x_1, . . . , x_n\}$ be a finite subset of M. Denote the element $(x_1, . . . , x_n)$ of $M^n$ by $x$. The A-module $M^n$ is semisimple and, by Proposition 2 of VIII, p. 79, its bicommutant coincides with the homotheties of the $A''_M$-module $M^n$. By Lemma 3, the A-submodule $Ax$ of $M^n$ is an $A''_M$-submodule of $M^n$. Let $a\in A$ be such that $(cx_1, . . . , cx_n)$ is equal to $ax$. Then $c$ coincides with $a_M$ on $\{x_1, . . . , x_n\}$, which implies condition (D).