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<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 ;<br/><b>assume </b><a NAME="E2:14_1"/>
not <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E3:14_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:14_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:14_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E8:14_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E9:14_1"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:14_1"/>
<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> <font color="Maroon">k</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/><b>hence </b><a NAME="E11:14_1"/>
<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> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><a NAME="E12:14_1"/>
<font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:14_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="binop_2.html#K24">*</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> ) )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1"/>
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="binop_2.html#K24">*</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_2_1_1"/>
not <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="binop_2.html#K24">*</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:14_2_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 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> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><a NAME="E4:14_2_1_2"/>
( <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 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> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:14_2_1_2"/>
not <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="binop_2.html#K24">*</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">n</font> is <a href="int_2.html#V1">prime</a> ) ) iff  not <font color="Olive">p</font> is <a href="int_2.html#V1">prime</a> )
 <b>by </b><i/>;<br/>


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