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therefore, to the commutative, associative, and distributive laws.

§4
The Exponentiation of Powers

By a "covering of the aggregate $N$ with elements of the aggregate $M$," or, more simply, by a "covering of $N$ with $M$," we understand a law by which with every element $n$ of $N$ a definite element of $M$ is bound up, where one and the same element of $M$ can come repeatedly into application. The element of $M$ bound up with $n$ is, in a way, a one-valued function of $n$, and may be denoted by $f(n)$; it is called a "covering function of $n$." The corresponding covering of $N$ will be called $f(N)$.

[487] Two coverings $f_{1}(N)$ and $f_{2}(N)$ are said to be equal if, and only if, for all elements $n$ of $N$ the equation

is fulfilled, so that if this equation does not subsist for even a single element $n=n_{0}$,$f_{1}(N)$ and $f_{2}(N)$ are characterized as different coverings of $N$. For example, if $m_{0}$ is a particular element of $M$, we may fix that, for all $n$'s

this law constitutes a particular covering of $N$ with $M$. Another kind of covering results if $m_{0}$ and $m_{1}$ are two different particular elements of $M$ and $n_{0}$ a particular element of $N$, from fixing that