Question 1009274
The idea is to assume ~C is true (line 3). Using the <a href = "https://www.blinn.edu/brazos/humanities/avoelkel/logic%20note%20sheet.pdf">rules of inference/replacement</a>, if we can lead to A v B somehow (line 25), then that proves ~C -> (A v B) is true.


<table border=1><tr><th colspan="2">Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td></td><td>(A -&gt; E) -&gt; (D v C)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>D -&gt; (~B -&gt; C)</td><td></td><td></td></tr><tr><td>:.</td><td></td><td>~C -&gt; (A v B)</td><td></td><td></td></tr><tr><td></td><td>3</td><td>~C</td><td></td><td>ACP</td></tr><tr><td></td><td>4</td><td>(D &amp; ~B) -&gt; C</td><td>2</td><td>EXP</td></tr><tr><td></td><td>5</td><td>~(D &amp; ~B)</td><td>4,3</td><td>MT</td></tr><tr><td></td><td>6</td><td>~D v ~~B</td><td>5</td><td>DM</td></tr><tr><td></td><td>7</td><td>~D v B</td><td>6</td><td>DN</td></tr><tr><td></td><td>8</td><td>D -&gt; B</td><td>7</td><td>MI</td></tr><tr><td></td><td>9</td><td>~B -&gt; ~D</td><td>8</td><td>Trans</td></tr><tr><td></td><td>10</td><td>~(A -&gt; E) v (D v C)</td><td>1</td><td>MI</td></tr><tr><td></td><td>11</td><td>~(~A v E) v (D v C)</td><td>10</td><td>MI</td></tr><tr><td></td><td>12</td><td>(~~A &amp; ~E) v (D v C)</td><td>11</td><td>DM</td></tr><tr><td></td><td>13</td><td>(A &amp; ~E) v (D v C)</td><td>12</td><td>DN</td></tr><tr><td></td><td>14</td><td>(D v C) v (A &amp; ~E)</td><td>13</td><td>Comm</td></tr><tr><td></td><td>15</td><td>[(D v C) v A] &amp; [(D v C) v ~E]</td><td>14</td><td>Dist</td></tr><tr><td></td><td>16</td><td>(D v C) v A</td><td>15</td><td>Simp</td></tr><tr><td></td><td>17</td><td>(C v D) v A</td><td>16</td><td>Comm</td></tr><tr><td></td><td>18</td><td>C v (D v A)</td><td>17</td><td>Assoc</td></tr><tr><td></td><td>19</td><td>D v A</td><td>18,3</td><td>DS</td></tr><tr><td></td><td>20</td><td>~~D v A</td><td>19</td><td>DN</td></tr><tr><td></td><td>21</td><td>~D -&gt; A</td><td>20</td><td>MI</td></tr><tr><td></td><td>22</td><td>~B -&gt; A</td><td>9,21</td><td>HS</td></tr><tr><td></td><td>23</td><td>~~B v A</td><td>22</td><td>MI</td></tr><tr><td></td><td>24</td><td>B v A</td><td>23</td><td>DN</td></tr><tr><td></td><td>25</td><td>A v B</td><td>24</td><td>Comm</td></tr><tr><td>26</td><td></td><td>~C -&gt; (A v B)</td><td>3-25</td><td>CP</td></tr></table>


Acroynyms/Abbreviations used


ACP = assumption for conditional proof
Assoc = associative property
Comm = commutation
CP = conditional proof
Dist = distribution
DM = de morgan's law
DN = double negation
DS = disjunctive syllogism
EXP = exportation
HS = hypothetical syllogism
MI = material implication
MT = modus tollens
Simp = simplification
Trans = transposition