Set of Condensation Points is Subset of Derivative

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A$ be a subset of $S$.

Then:
 * $A^0 \subseteq A'$

where
 * $A^0$ denotes the set of condensation points of $A$
 * $A'$ denotes the derivative of $A$

Proof
Let $x \in A^0$.

By definition of set of condensation points:
 * $x$ is condensation point of $A$

By definition of condensation point:
 * $x$ is limit point of $A$

By definition of derived set:
 * $x \in A'$