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B. If two aggregates $M$ and $N$ are such that $M$ is equivalent to a part $N_{1}$ of $N$ and $N$ to a part $M_{1}$ of $M$, then $M$ and $N$ are equivalent;

C. If $M_{1}$ is a part of an aggregate $M$, $M_{2}$ is a part of the aggregate $M_{1}$, and if the aggregates $M$ and $M_{2}$ are equivalent, then $M_{1}$ is equivalent to both $M$ and $M_{2}$;

D. If, with two aggregates $M$ and $N$, $N$ is equivalent neither to $M$ nor to a part of $M$, there is a part $N_{1}$ of $N$ that is equivalent to $M$;

E. If two aggregates $M$ and $N$ are not equivalent, and there is a part $N_{1}$ of $N$ that is equivalent to $M$, then no part of $M$ is equivalent to $N$.

The Addition and Multiplication of Powers

The union of two aggregates $M$ and $N$ which have no common elements was denoted in § 1, (2), by $(M,N)$. We call it the "union-aggregate (Vereinigungsmenge) of $M$ and $N$."

If $M'$ and $N'$ are two other aggregates without common elements, and if $M\sim M'$ and $N\sim N'$, we saw that we have

$(M,N)\sim (M',N')$.

Hence the cardinal number of $(M,N)$ only depends upon the cardinal numbers ${\overline {\overline {M}}}={\mathfrak {a}}$ and ${\overline {\overline {N}}}={\mathfrak {b}}$.

This leads to the definition of the sum of a and b. We put