Exercise 5, § 3

Tag 03L7, permanent. Cite it as [Bourbaki, Tag 03L7].

Theory of Sets, III, 230

Let $(\lambda_\iota)_{\iota \in I}$ be a family of order-types ( $§ 2$ , Exercise 13), indexed by an ordered set I. Show that

\mathrm{Card} \left( \sum_{\iota \in I} \lambda_\iota \right) = \sum_{\iota \in I} \mathrm{Card}\ (\lambda_\iota)

and that, if I is well-ordered,

\mathrm{Card} \left( \mathop{\mathbf{P}}_{\iota \in I} \lambda_\iota \right) = \mathop{\mathbf{P}}_{\iota \in I} \mathrm{Card}\ (\lambda_\iota).

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