Lemma 2

Tag 05PT, permanent. Cite it as [Bourbaki, Tag 05PT].

Let $M, N$ be two supplementary submodules in $F$ and $i : M \to F, j : N \to F$ the canonical injections. The diagram

\begin{array}{ccc} T(M) \oplus T(N) & \xrightarrow{\nu_M \oplus \nu_N} & T'(M) \oplus T'(N) \\ \downarrow T(i) + T(j) & & \downarrow T'(i) + T'(j) \\ T(F) & \xrightarrow{\nu_F} & T'(F) \end{array}

is commutative and the vertical arrows are bijective.

The commutativity follows from Lemma 1, the other assertions from § 1, no. 6, Corollary 2 to Proposition 6 and § 3, no. 7, Proposition 7.

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

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