Fill in the appropriate blanks to complete the derivation. Provide both the rule justifying the line and line number$($s$)$ from which it is derived.

A sub$-$derivation is indicated by a bold vertical line $(|)$ after the number, and additional lines are added the more sub$-$proofs we have. So $|$ would indicate one sub$-$proof deep, $||$ would indicate we are two deep, $|||$ would indicate three sub$-$derivations deep, and so on$...$

This picture should help sort out any confusion.

picture of derivation below

If the special logical symbols are not available or won't display properly in questions and$/$or your answers, the following may be used:

~ for

$>$ for

$<>$ for

$+$ for

v and & are on the normal keyboard layout

All symbolic notation $($except as noted above$)$ must be exactly as used in OLI: Logic & Proofs.

All derivation, tree, and other rules must be applied exactly as presented in OLI: Logic & Proofs.

All derivation, tree, and other rule names must be exactly as given in OLI: Logic & Proofs.

You do not need to designate R$($right$)$ or L$($left$)$ on rules.

$1.(($O $->$ N$)->$P$)$ premise

$2.($O & $($P v N$))$ premise

$3.($P v N$)$

$4.$ O

$5.|$N assume

$6.||$ P assume

$7.||$P

$8.||($O $->$ N$)$

$9.||($O $->$ N$)$

$10.||$ N

$11.||$

Note: This ends one subproof

$12.||$ N assume

$13.||$

$14.|$

$15.$ N