The proof goes as follows:

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> Let $A$, $B$ be basis for $F$. For a fixed prime $p$, it is easy to see that the quotient group $F/pF$ is a vector space over $\Bbb Z /p \Bbb Z$ and that the cosets $\{a + pF: a \in A\}$ form a basis. And $\dim F/pF =\text{card } A = \text{card } B$.
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I don't see how $F/pF$ is a vector space over $\Bbb Z / p\Bbb Z$. I understand that $\Bbb Z / p \Bbb Z = \{ c + p \Bbb Z : c \in \Bbb Z\}$, where $p \Bbb Z = \{ p\*z : z \in \Bbb Z\}$ (multiplication operation), but how is $pF$ defined since $F/pF = \{ h + pF: h \in F\}$ and free abelian groups are defined in terms of addition?