Example 2

Tag 03VF, permanent. Cite it as [Bourbaki, Tag 03VF].

Theory of Sets, chapter IV, § 1. Structures and isomorphisms

The species of vector space structures over $\mathbf{C}$ (resp. $\mathbf{R}$ ) has a principal base set $E$ , an auxiliary base set equal to $\mathbf{C}$ (resp. $\mathbf{R}$ ), and typical characterization

s_1\in\mathfrak{P}((E\times E)\times E)\qquad\text{and}\qquad s_2\in\mathfrak{P}((\mathbf{C}\times E)\times E)

(resp. $s_1\in\mathfrak{P}((E\times E)\times E)$ and $s_2\in\mathfrak{P}((\mathbf{R}\times E)\times E)$ ).

The pair $(s_1,s_2\cap((\mathbf{R}\times E)\times E))$ is a procedure of deduction of a vector space structure over $\mathbf{R}$ from a vector space structure over $\mathbf{C}$ (“restriction of the field of scalars to $\mathbf{R}$ ”). \*

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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