Rule of Addition/Sequent Form/Formulation 2/Proof by Truth Table

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives are $T$ for all boolean interpretations.

$\begin{array}{|c|c|ccccc|ccccc|} \hline p & q & p & \implies & (p & \lor & q) & q & \implies & (p & \lor & q) \\ \hline F & F & F & T & F & F & F & F & T & F & F & F \\ F & T & F & T & F & T & T & T & T & F & T & T \\ T & F & T & T & T & T & F & F & T & T & T & F \\ T & T & T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$