# Absorption Laws (Set Theory)/Union with Intersection/Proof 2

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## Theorem

- $S \cup \paren {S \cap T} = S$

## Proof

\(\displaystyle x\) | \(\in\) | \(\displaystyle S \cup \paren {S \cap T}\) | |||||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle x\) | \(\in\) | \(\displaystyle S \lor \paren {x \in S \land x \in T}\) | Definition of Set Intersection and Definition of Set Union | |||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle x\) | \(\in\) | \(\displaystyle S\) | Disjunction Absorbs Conjunction |

$\blacksquare$