Rule of Addition/Sequent Form/Proof by Truth Table

Theorem

 * $(1): \quad p \vdash p \lor q$
 * $(2): \quad p \vdash q \lor p$

Proof
We apply the Method of Truth Tables.

$\begin{array}{|c|c||ccc|} \hline p & q & p & \lor & q\\ \hline F & F & F & F & F \\ F & T & F & T & T \\ T & F & T & T & F \\ T & T & T & T & T \\ \hline \end{array}$

As can be seen, whenever either $p$ or $q$ (or both) are true, then so is $p \lor q$.