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<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="E1:35"/><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="xxreal_0.html#R1">&lt;</a> 29 &amp; <font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a> )
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E39</font></i>: 
( 1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 28 <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( ( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 5 ) or ( 5 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 27 <a href="binop_2.html#K23">+</a> 1 ) )
 </span><b>by </b><i><a class="ref" href="nat_4.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/>
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 5 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 7 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 13 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 17 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 19 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 23 )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green">E40</font></i>: 
( 5 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 27 <a href="binop_2.html#K23">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_2_1"/>
( ( 5 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 12 <a href="binop_2.html#K23">+</a> 1 ) or ( 13 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 20 <a href="binop_2.html#K23">+</a> 1 ) or ( 21 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 27 <a href="binop_2.html#K23">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_1"><b>suppose </b></a><a NAME="E1:35_1_2_1_1"/>
( 5 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 12 <a href="binop_2.html#K23">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:35_1_2_1_1"/><b>then </b>
( ( 5 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 <a href="binop_2.html#K23">+</a> 1 ) or ( 6 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 6 <a href="binop_2.html#K23">+</a> 1 ) or ( 7 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 7 <a href="binop_2.html#K23">+</a> 1 ) or ( 8 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 8 <a href="binop_2.html#K23">+</a> 1 ) or ( 9 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 9 <a href="binop_2.html#K23">+</a> 1 ) or ( 10 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 10 <a href="binop_2.html#K23">+</a> 1 ) or ( 11 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 11 <a href="binop_2.html#K23">+</a> 1 ) or ( 12 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 12 <a href="binop_2.html#K23">+</a> 1 ) )
 ;<br/><b>hence </b><a NAME="E3:35_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 7 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 13 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 17 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 19 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 23 )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_2"><b>suppose </b></a><a NAME="E1:35_1_2_1_2"/>
( 13 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 20 <a href="binop_2.html#K23">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:35_1_2_1_2"/><b>then </b>
( ( 13 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 13 <a href="binop_2.html#K23">+</a> 1 ) or ( 14 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 14 <a href="binop_2.html#K23">+</a> 1 ) or ( 15 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 15 <a href="binop_2.html#K23">+</a> 1 ) or ( 16 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 16 <a href="binop_2.html#K23">+</a> 1 ) or ( 17 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 17 <a href="binop_2.html#K23">+</a> 1 ) or ( 18 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 18 <a href="binop_2.html#K23">+</a> 1 ) or ( 19 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 19 <a href="binop_2.html#K23">+</a> 1 ) or ( 20 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 20 <a href="binop_2.html#K23">+</a> 1 ) )
 ;<br/><b>hence </b><a NAME="E3:35_1_2_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 7 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 13 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 17 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 19 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 23 )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_3"><b>suppose </b></a><a NAME="E1:35_1_2_1_3"/>
( 21 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 27 <a href="binop_2.html#K23">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:35_1_2_1_3"/><b>then </b>
( ( 21 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 21 <a href="binop_2.html#K23">+</a> 1 ) or ( 22 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 22 <a href="binop_2.html#K23">+</a> 1 ) or ( 23 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 23 <a href="binop_2.html#K23">+</a> 1 ) or ( 24 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 24 <a href="binop_2.html#K23">+</a> 1 ) or ( 25 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 25 <a href="binop_2.html#K23">+</a> 1 ) or ( 26 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 26 <a href="binop_2.html#K23">+</a> 1 ) or ( 27 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 27 <a href="binop_2.html#K23">+</a> 1 ) )
 ;<br/><b>hence </b><a NAME="E3:35_1_2_1_3"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 7 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 13 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 17 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 19 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 23 )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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