Product of Subset with Union/Proof 2

Proof
Consider the relation $\mathcal R \subseteq G \times G$ defined as:


 * $\forall g, h \in G: \tuple {g, h} \in \mathcal R \iff \exists g \in X$

Then:
 * $\forall S \subseteq G: X \circ S = \map {\mathcal R} S$

Then:

Next, consider the relation $\mathcal R \subseteq G \times G$ defined as:


 * $\forall g, h \in G: \tuple {g, h} \in \mathcal R \iff \exists h \in X$

Then:
 * $\forall S \subseteq G: S \circ X = \map {\mathcal R} S$

Then: