Question 1169573
Here's one way to do the derivation. I'm using arrow symbols in place of horseshoe symbols. 


<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td><td>Lines Used</td><td>Reason</td></tr><tr><td>1</td><td>~N -> (~R -> C)</td><td></td><td></td></tr><tr><td>2</td><td>R -> N</td><td></td><td></td></tr><tr><td>3</td><td>~C</td><td></td><td></td></tr><tr><td>:.</td><td>N</td><td></td><td></td></tr><tr><td>4</td><td>(~N & ~R) -> C</td><td>1</td><td>Exportation</td></tr><tr><td>5</td><td>~(~N & ~R)</td><td>4,3</td><td>Modus Tollens</td></tr><tr><td>6</td><td>~~N v ~~R</td><td>5</td><td>De Morgan’s Law</td></tr><tr><td>7</td><td>~~N v R</td><td>6</td><td>Double Negation</td></tr><tr><td>8</td><td>~N -> R</td><td>7</td><td>Material Implication</td></tr><tr><td>9</td><td>~N -> N</td><td>8,2</td><td>Hypothetical Syllogism</td></tr><tr><td>10</td><td>~~N v N</td><td>9</td><td>Material Implication</td></tr><tr><td>11</td><td>N v N</td><td>10</td><td>Double Negation</td></tr><tr><td>12</td><td>N</td><td>11</td><td>Tautology</td></tr></table>