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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 631 &amp; <font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a> )
 ;<br/><a NAME="E2:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 315</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E3:43_1"/><b>then </b><i><font color="Green">E39</font></i>: 
not 2 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E4:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> 210</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E5:43_1"/><b>then </b><i><font color="Green">E40</font></i>: 
not 3 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E6:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="binop_2.html#K24">*</a> 126</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E7:43_1"/><b>then </b><i><font color="Green">E46</font></i>: 
not 5 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E8:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">7 <a href="binop_2.html#K24">*</a> 90</span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E9:43_1"/><b>then </b><i><font color="Green">E47</font></i>: 
not 7 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E10:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">11 <a href="binop_2.html#K24">*</a> 57</span>)</span> <a href="binop_2.html#K23">+</a> 4
 ;<br/><a NAME="E11:43_1"/><b>then </b><i><font color="Green">E48</font></i>: 
not 11 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E12:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">13 <a href="binop_2.html#K24">*</a> 48</span>)</span> <a href="binop_2.html#K23">+</a> 7
 ;<br/><a NAME="E13:43_1"/><b>then </b><i><font color="Green">E53</font></i>: 
not 13 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E14:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">17 <a href="binop_2.html#K24">*</a> 37</span>)</span> <a href="binop_2.html#K23">+</a> 2
 ;<br/><a NAME="E15:43_1"/><b>then </b><i><font color="Green">E54</font></i>: 
not 17 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E16:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">19 <a href="binop_2.html#K24">*</a> 33</span>)</span> <a href="binop_2.html#K23">+</a> 4
 ;<br/><a NAME="E17:43_1"/><b>then </b><i><font color="Green">E55</font></i>: 
not 19 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><a NAME="E18:43_1"/>
631 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">23 <a href="binop_2.html#K24">*</a> 27</span>)</span> <a href="binop_2.html#K23">+</a> 10
 ;<br/><a NAME="E19:43_1"/><b>then </b>
not 23 <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E20:43_1"/>
not <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> 631
 <b>by </b><i><a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, , , , <a class="txt" href="nat_4.html#E7:43_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="nat_4.html#E9:43_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nat_4.html#E11:43_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_4.html#T4" target="_self">Th4</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43"/>
631 is <a href="int_2.html#V1">prime</a>
 <b>by </b><i/>;<br/>


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