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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 317 &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> )
 ;<br/><a NAME="E2:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 158</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E3:42_1"/><b>then </b><i><font color="Green">E42</font></i>: 
not 2 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> 105</span>)</span> <a href="binop_2.html#K23">+</a> 2
 ;<br/><a NAME="E5:42_1"/><b>then </b><i><font color="Green">E43</font></i>: 
not 3 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E6:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="binop_2.html#K24">*</a> 63</span>)</span> <a href="binop_2.html#K23">+</a> 2
 ;<br/><a NAME="E7:42_1"/><b>then </b><i><font color="Green">E44</font></i>: 
not 5 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E8:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">7 <a href="binop_2.html#K24">*</a> 45</span>)</span> <a href="binop_2.html#K23">+</a> 2
 ;<br/><a NAME="E9:42_1"/><b>then </b><i><font color="Green">E45</font></i>: 
not 7 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E10:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">11 <a href="binop_2.html#K24">*</a> 28</span>)</span> <a href="binop_2.html#K23">+</a> 9
 ;<br/><a NAME="E11:42_1"/><b>then </b><i><font color="Green">E46</font></i>: 
not 11 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E12:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">13 <a href="binop_2.html#K24">*</a> 24</span>)</span> <a href="binop_2.html#K23">+</a> 5
 ;<br/><a NAME="E13:42_1"/><b>then </b><i><font color="Green">E47</font></i>: 
not 13 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E14:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">17 <a href="binop_2.html#K24">*</a> 18</span>)</span> <a href="binop_2.html#K23">+</a> 11
 ;<br/><a NAME="E15:42_1"/><b>then </b><i><font color="Green">E48</font></i>: 
not 17 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E16:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">19 <a href="binop_2.html#K24">*</a> 16</span>)</span> <a href="binop_2.html#K23">+</a> 13
 ;<br/><a NAME="E17:42_1"/><b>then </b><i><font color="Green">E49</font></i>: 
not 19 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E18:42_1"/>
317 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">23 <a href="binop_2.html#K24">*</a> 13</span>)</span> <a href="binop_2.html#K23">+</a> 18
 ;<br/><a NAME="E19:42_1"/><b>then </b>
not 23 <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E20:42_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> 317
 <b>by </b><i><a class="txt" href="nat_4.html#E1:42_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nat_4.html#E3:42_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nat_4.html#E5:42_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nat_4.html#E7:42_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nat_4.html#E9:42_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nat_4.html#E11:42_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="nat_4.html#E13:42_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nat_4.html#E15:42_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nat_4.html#E17:42_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_4.html#E36"><i><font color="Green">Lemma34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:42"/>
317 is <a href="int_2.html#V1">prime</a>
 <b>by </b><i><a class="ref" href="nat_4.html#T14" target="_self">Th14</a></i>;<br/>


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