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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> , <font color="Maroon">f</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">d</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">d</font> &amp; <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">q</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">q</font> is <a href="int_2.html#V1">prime</a> )
 ;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:26"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font>;<br/>
<a NAME="E4:26"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E47</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:26"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">q1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font> &amp; <font color="Olive">q1</font> is <a href="int_2.html#V1">prime</a> holds <br/> ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <font color="Olive">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">q1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font>
 ;<br/>

<b>assume </b><a NAME="E2:26_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">q1</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<a NAME="E3:26_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:26_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:26_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">a</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="E5:26_1"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E6:26_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , , <a class="ref" href="nat_4.html#T1" target="_self">Th1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:26"/>
<font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a>
 <b>by </b><i>, , , </i>;<br/>


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