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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_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="E1:40_1"/><i><font color="Green">E39</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> 139 &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> )
 ;<br/><a NAME="E2:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 69</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E3:40_1"/><b>then </b><i><font color="Green">E40</font></i>: 
not 2 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> 46</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E5:40_1"/><b>then </b><i><font color="Green">E41</font></i>: 
not 3 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E6:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="binop_2.html#K24">*</a> 27</span>)</span> <a href="binop_2.html#K23">+</a> 4
 ;<br/><a NAME="E7:40_1"/><b>then </b><i><font color="Green">E42</font></i>: 
not 5 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E8:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">7 <a href="binop_2.html#K24">*</a> 19</span>)</span> <a href="binop_2.html#K23">+</a> 6
 ;<br/><a NAME="E9:40_1"/><b>then </b><i><font color="Green">E43</font></i>: 
not 7 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E10:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">11 <a href="binop_2.html#K24">*</a> 12</span>)</span> <a href="binop_2.html#K23">+</a> 7
 ;<br/><a NAME="E11:40_1"/><b>then </b><i><font color="Green">E44</font></i>: 
not 11 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E12:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">13 <a href="binop_2.html#K24">*</a> 10</span>)</span> <a href="binop_2.html#K23">+</a> 9
 ;<br/><a NAME="E13:40_1"/><b>then </b><i><font color="Green">E45</font></i>: 
not 13 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E14:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">17 <a href="binop_2.html#K24">*</a> 8</span>)</span> <a href="binop_2.html#K23">+</a> 3
 ;<br/><a NAME="E15:40_1"/><b>then </b><i><font color="Green">E46</font></i>: 
not 17 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E16:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">19 <a href="binop_2.html#K24">*</a> 7</span>)</span> <a href="binop_2.html#K23">+</a> 6
 ;<br/><a NAME="E17:40_1"/><b>then </b><i><font color="Green">E47</font></i>: 
not 19 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E18:40_1"/>
139 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">23 <a href="binop_2.html#K24">*</a> 6</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E19:40_1"/><b>then </b>
not 23 <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="ref" href="nat_4.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E20:40_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> 139
 <b>by </b><i><a class="txt" href="nat_4.html#E1:40_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nat_4.html#E3:40_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nat_4.html#E5:40_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nat_4.html#E7:40_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nat_4.html#E9:40_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nat_4.html#E11:40_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nat_4.html#E13:40_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nat_4.html#E15:40_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="nat_4.html#E17:40_1"><i><font color="Green">E47</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="E2:40"/>
139 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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