Corollary

Let $f$ be an injection of A into B. For every subset X of A, we have $f\langle \mathrm{A} - \mathrm{X} \rangle = f\langle \mathrm{A} \rangle - f\langle \mathrm{X} \rangle$.

Writing $f = i \circ g$, where $i$ is the canonical injection of $f\langle \mathrm{A} \rangle$ into B, we reduce the Corollary to Proposition 6 applied to $\overset{-1}{g}$.

¶ The intersection $\mathrm{X} \cap \mathrm{A}$ is sometimes called the trace of X on A. If $\mathfrak{F}$ is a family of sets, the set of traces on A of the sets belonging to $\mathfrak{F}$ is called the trace of $\mathfrak{F}$ on A.