Let G 1 , … , G n G_1, \ldots, G_n G 1 , … , G n G = G 1 × ⋯ × G n G = G_1 \times \cdots \times G_n G = G 1 × ⋯ × G n T ( ∞ ) ( G ) \mathscr{T}^{(\infty)}(G) T ( ∞ ) ( G ) T ( ∞ ) ( G 1 ) ⊗ ⋯ ⊗ T ( ∞ ) ( G n ) \mathscr{T}^{(\infty)}(G_1) \otimes \cdots \otimes \mathscr{T}^{(\infty)}(G_n) T ( ∞ ) ( G 1 ) ⊗ ⋯ ⊗ T ( ∞ ) ( G n ) T ( ∞ ) ( G ) \mathscr{T}^{(\infty)}(G) T ( ∞ ) ( G ) T ( ∞ ) ( G 1 ) , … , T ( ∞ ) ( G n ) \mathscr{T}^{(\infty)}(G_1), \ldots, \mathscr{T}^{(\infty)}(G_n) T ( ∞ ) ( G 1 ) , … , T ( ∞ ) ( G n ) t i ∈ T ( ∞ ) ( G i ) t_i \in \mathscr{T}^{(\infty)}(G_i) t i ∈ T ( ∞ ) ( G i ) i = 1 , … , n i = 1, \ldots, n i = 1 , … , n
( t 1 ⊗ ⋯ ⊗ t n ) ∨ = t 1 ∨ ⊗ ⋯ ⊗ t n ∨ .
(t_1 \otimes \cdots \otimes t_n)^\vee = t_1^\vee \otimes \cdots \otimes t_n^\vee.
( t 1 ⊗ ⋯ ⊗ t n ) ∨ = t 1 ∨ ⊗ ⋯ ⊗ t n ∨ . It suffices to consider the case n = 2 n = 2 n = 2 t 1 , t 1 ′ t_1, t_1' t 1 , t 1 ′ T ( ∞ ) ( G 1 ) \mathscr{T}^{(\infty)}(G_1) T ( ∞ ) ( G 1 ) t 2 , t 2 ′ t_2, t_2' t 2 , t 2 ′ T ( ∞ ) ( G 2 ) \mathscr{T}^{(\infty)}(G_2) T ( ∞ ) ( G 2 ) ( t 1 ⊗ t 2 ) ∗ ( t 1 ′ ⊗ t 2 ′ ) = ( t 1 ∗ t 1 ′ ) ⊗ ( t 2 ∗ t 2 ′ ) (t_1 \otimes t_2) * (t_1' \otimes t_2') = (t_1 * t_1') \otimes (t_2 * t_2') ( t 1 ⊗ t 2 ) ∗ ( t 1 ′ ⊗ t 2 ′ ) = ( t 1 ∗ t 1 ′ ) ⊗ ( t 2 ∗ t 2 ′ ) ( t 1 ⊗ t 2 ) ∨ = t 1 ∨ ⊗ t 2 ∨ (t_1 \otimes t_2)^\vee = t_1^\vee \otimes t_2^\vee ( t 1 ⊗ t 2 ) ∨ = t 1 ∨ ⊗ t 2 ∨
G × G → s G × G ↓ n ↓ m G G
\begin{array}{ccc}
G \times G & \xrightarrow{s} & G \times G \\
\downarrow n & & \downarrow m \\
G & & G
\end{array}
G × G ↓ n G s G × G ↓ m G where m ( ( x 1 , x 2 ) , ( x 1 ′ , x 2 ′ ) ) = ( x 1 x 1 ′ , x 2 x 2 ′ ) m((x_1, x_2), (x_1', x_2')) = (x_1 x_1', x_2 x_2') m (( x 1 , x 2 ) , ( x 1 ′ , x 2 ′ )) = ( x 1 x 1 ′ , x 2 x 2 ′ )
n ( ( x 1 , x 2 ) , ( x 1 ′ , x 2 ′ ) ) = ( ( x 1 , x 1 ′ ) , ( x 2 , x 2 ′ ) ) ,
n((x_1, x_2), (x_1', x_2')) = ((x_1, x_1'), (x_2, x_2')),
n (( x 1 , x 2 ) , ( x 1 ′ , x 2 ′ )) = (( x 1 , x 1 ′ ) , ( x 2 , x 2 ′ )) , p 1 ( x 1 , x 1 ′ ) = x 1 x 1 ′ p_1(x_1, x_1') = x_1 x_1' p 1 ( x 1 , x 1 ′ ) = x 1 x 1 ′ p 2 ( x 2 , x 2 ′ ) = x 2 x 2 ′ p_2(x_2, x_2') = x_2 x_2' p 2 ( x 2 , x 2 ′ ) = x 2 x 2 ′
m ∗ ( ( t 1 ⊗ t 2 ) ⊗ ( t 1 ′ ⊗ t 2 ′ ) ) = ( p 1 ⊗ p 2 ) ∗ ( n ∗ ( ( t 1 ⊗ t 2 ) ⊗ ( t 1 ′ ⊗ t 2 ′ ) ) ) ,
m_*((t_1 \otimes t_2) \otimes (t_1' \otimes t_2')) = (p_1 \otimes p_2)_*(n_*((t_1 \otimes t_2) \otimes (t_1' \otimes t_2'))),
m ∗ (( t 1 ⊗ t 2 ) ⊗ ( t 1 ′ ⊗ t 2 ′ )) = ( p 1 ⊗ p 2 ) ∗ ( n ∗ (( t 1 ⊗ t 2 ) ⊗ ( t 1 ′ ⊗ t 2 ′ ))) , that is
( t 1 ⊗ t 2 ) ∗ ( t 1 ′ ⊗ t 2 ′ ) = ( p 1 ⊗ p 2 ) ∗ ( ( t 1 ⊗ t 1 ′ ) ⊗ ( t 2 ⊗ t 2 ′ ) ) = p 1 ∗ ( t 1 ⊗ t 1 ′ ) ⊗ p 2 ∗ ( t 2 ⊗ t 2 ′ ) = ( t 1 ∗ t 1 ′ ) ⊗ ( t 2 ∗ t 2 ′ ) .
\begin{align*}
(t_1 \otimes t_2) * (t_1' \otimes t_2') &= (p_1 \otimes p_2)_*((t_1 \otimes t_1') \otimes (t_2 \otimes t_2')) \\
&= p_1*(t_1 \otimes t_1') \otimes p_2*(t_2 \otimes t_2') \\
&= (t_1 * t_1') \otimes (t_2 * t_2').
\end{align*}
( t 1 ⊗ t 2 ) ∗ ( t 1 ′ ⊗ t 2 ′ ) = ( p 1 ⊗ p 2 ) ∗ (( t 1 ⊗ t 1 ′ ) ⊗ ( t 2 ⊗ t 2 ′ )) = p 1 ∗ ( t 1 ⊗ t 1 ′ ) ⊗ p 2 ∗ ( t 2 ⊗ t 2 ′ ) = ( t 1 ∗ t 1 ′ ) ⊗ ( t 2 ∗ t 2 ′ ) . It is seen analogously that ( t 1 ⊗ t 2 ) ∨ = t 1 ∨ ⊗ t 2 ∨ (t_1 \otimes t_2)^\vee = t_1^\vee \otimes t_2^\vee ( t 1 ⊗ t 2 ) ∨ = t 1 ∨ ⊗ t 2 ∨
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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