Derivative of Derivative is Subset of Derivative in T1 Space

Theorem
Let $T = \struct {S, \tau}$ be a $T_1$ topological space.

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

Then
 * $A'' \subseteq A'$

where
 * $A'$ denotes the derivative of $A$

Proof
Let:
 * $(1): \quad x \in A''$.

$x \notin A'$.

Then by Characterization of Derivative by Open Sets there exists an open subset $G$ of $T$ such that:
 * $(2): \quad x \in G$ and
 * $(3): \quad \lnot \exists y: y \in A \cap G \land x \ne y$.

By definition of $T_1$ space:
 * $\left\{{x}\right\}$ is closed.

Then by Open Set minus Closed Set is Open:
 * $(4): \quad G \setminus \set x$ is open.

By $(1)$, $(2)$, and Characterization of Derivative by Open Sets there exists a point $y$ of $T$ such that:
 * $(5): \quad y \in A' \cap G$ and
 * $(6): \quad x \ne y$.

Then by definition of intersection:
 * $y \in A'$.

Then by $(6)$ and definition of set difference:
 * $(7): \quad y \in A' \setminus \set x$.

By definition of intersection and $(5)$:
 * $y \in G$.

By $(6)$ and definition of singleton:
 * $y \notin \set x$

Then by definition of set difference:
 * $(8): \quad y \in G \setminus \set x$

We will prove:
 * $(9): \quad G \cap \paren {A \setminus \set x} = \O$


 * $G \cap \paren {A \setminus \set x} \ne \O$.
 * $G \cap \paren {A \setminus \set x} \ne \O$.


 * Then by definition of the empty set there exists $g$ such that:
 * $g \in G \cap \paren {A \setminus \set x}$
 * Hence by definition of intersection:
 * $g \in G$ and
 * $g \in A \setminus \set x$.
 * Then by definition of set difference:
 * $g \in A$
 * Hence by definition of intersection:
 * $g \in A \cap G$
 * Then by $(3)$:
 * $x = g$
 * Hence this by definition of singleton contadicts with $g \notin \set x$ obtained by definition of set difference.
 * Thus $G \cap \paren {A \setminus \set x} = \O$.

Define $U = G \setminus \set x$ as an open set by $(4)$.

By $(5)$ and definition of set difference:
 * $y \in A'$

Then by $(8)$ and Characterization of Derivative by Open Sets there exists a point $q$ of $T$ such that
 * $(10): \quad q \in A \cap U$ and
 * $(11): \quad y \ne q$.

By $(10)$ and definition of intersection:
 * $q \in A$

By $(11)$ and definition of singleton:
 * $q \notin \set y$

Then by definition of set difference
 * $(12): \quad q \in A \setminus \set y$.

By definition of intersection:
 * $q \in U$.

Then by $(12)$ and by definition of set difference
 * $q \ne x$ and $q \in A$.

Then by definition of set difference:
 * $q \in A \setminus \set x$

and
 * $q \in G$.

Hence this contradicts with $(9)$ by definition of intersection.

Thus the result by Proof by Contradiction.