Lemma 2

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

Theory of Sets, chapter I, Appendix 0. Characterization of terms and relations

Every balanced word $A$ can be put in the form

A=fA_1A_2\ldots A_p,

where the $A_i$ are balanced and $n(f)=p$ .

Let $f$ be the initial sign of $A$ . By Lemma 1, $A$ can be written as

fA_1A_2\ldots A_p,

where the $A_i$ are balanced : we need only define $A_i$ inductively as the balanced segment of $A$ which begins at the $k(i)$ th place, where

k(i)=2+\sum_{j<i}l(A_j).

Moreover, we have

1+l(A_1)+\ldots+l(A_p)=l(A)=n(A)+1

=n(f)+n(A_1)+\ldots+n(A_p)+1

=n(f)+(l(A_1)-1)+\ldots+(l(A_p)-1)+1,

from which it follows that $n(f)=p$ .

¶ Now that the two lemmas have been proved, it is obvious by induction on the length of $A$ that every balanced word $A$ is significant, by reason of Lemma 2 and Proposition 1.

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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Lemma 2

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