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88 and sufficient condition for the equality of their cardinal numbers.

[483] In fact, according to the above definition of power, the cardinal number $$\overline\overline{M}$$ remains unaltered if in the place of each of one or many or even all elements $$m$$ of $$M$$ other things are substituted. If, now, $$M\sim N$$, there is a law of co-ordination by means of which $$M$$ and $$N$$ are uniquely and reciprocally referred to one another; and by it to the element $$m$$ of $$M$$ corresponds the element $$n$$ of $$N$$. Then we can imagine, in the place of every element $$m$$ of $$M$$, the corresponding element $$n$$ of $$N$$ substituted, and, in this way, $$M$$ transforms into $$N$$ without alteration of cardinal number. Consequently

The converse of the theorem results from the remark that between the elements of $$M$$ and the different units of its cardinal number $$\overline\overline{M}$$ a reciprocally univocal (or bi-univocal) relation of correspondence subsists. For, as we saw, $$\overline\overline{M}$$ grows, so to speak, out of $$M$$ in such a way that from every element $$m$$ of $$M$$ a special unit of $$M$$ arises. Thus we can say that

In the same way $$N\sim\overline\overline{N}$$. If then $$\overline\overline{M} = \overline\overline{N}$$, we have, by (6), $$M\sim N$$.

We will mention the following theorem, which results immediately from the conception of