Set of Closed Subsets of Power Structure of Entropic Structure is Closed

Theorem
Let $\struct {S, \odot}$ be a magma.

Let $\struct {S, \odot}$ be an entropic structure.

Let $\struct {\powerset S, \odot_\PP}$ be the power structure of $\struct {S, \odot}$.

Let $\TT$ be the set of all submagmas of $\struct {S, \odot}$.

Then the algebraic structure $\struct {\TT, \odot_\PP}$ is a submagma of $\struct {\powerset S, \odot_\PP}$ which is itself an entropic structure.

Proof
Recall the definition of subset product:
 * $A \odot_\PP B = \set {a \odot b: \tuple {a, b} \in A \times B}$

First we show that:
 * $\forall A, B \in \TT: A \odot_\PP B \in \TT$

Let $A$ and $B$ be arbitrary elements of $\TT$.

Let $a$ and $c$ be arbitrary elements of $A$.

Let $b$ and $d$ be arbitrary elements of $B$.

Then we have:

Then:

That is:
 * $A \odot_\PP B$ is closed in $S$.

As $A$ and $B$ are arbitrary, it follows that $\struct {\TT, \odot_\PP}$ is closed in $\powerset S$.

Then we need to show that:
 * $\forall A, B, C, D \in \TT: \paren {A \odot_\PP B} \odot_\PP \paren {C \odot_\PP D} = \paren {A \odot_\PP C} \odot_\PP \paren {B \odot_\PP D}$

demonstrating the entropic nature of $\struct {\TT, \odot_\PP}$.

We have:

and the result follows.