# Rule of Addition/Sequent Form/Formulation 2/Form 1

## Theorem

$\vdash p \implies \left({p \lor q}\right)$

## Proof 1

By the tableau method of natural deduction:

$p \implies \paren {p \lor q}$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Premise (None)
2 1 $p \lor q$ Rule of Addition: $\lor \II_1$ 1
3 $p \implies \paren {p \lor q}$ Rule of Implication: $\implies \mathcal I$ 1 – 3 Assumption 1 has been discharged

$\blacksquare$

## Proof 2

This proof is derived in the context of the following proof system: Instance 2 of the Hilbert-style systems.

By the tableau method:

$p \implies \left({p \lor q}\right)$
Line Pool Formula Rule Depends upon Notes
1 $q \implies \paren{ p \lor q }$ Axiom $A2$
2 $p \implies \paren{ q \lor p }$ Rule $RST \, 1$ 1 $p \,/\, q, q \,/\, p$
3 $\paren{ p \lor q } \implies \paren{ q \lor p }$ Axiom $A3$
4 $\paren{ q \lor p } \implies \paren{ p \lor q }$ Rule $RST \, 1$ 3 $p \,/\, q, q \,/\, p$
5 $p \implies \paren{ p \lor q }$ Hypothetical Syllogism 2,4

$\blacksquare$