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94 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 (1) 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 (1) 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