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sary 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

${\overline {\overline {M}}}={\overline {\overline {N}}}$.

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

(9)	$M\sim {\overline {\overline {M}}}$

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 equival-