<?xml version="1.0"?>
<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> , <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">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25"/><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> <font color="Maroon">f</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">f</font> &amp; <font color="Olive">q</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">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>assume </b><a NAME="E3:25"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<a NAME="E4:25"/>
<font color="Maroon">f</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:25"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <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/>
<b>then consider </b><font color="Maroon">p</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="E7:25"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:25"/><i><font color="Green">E47</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="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Olive">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Olive">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="E1:25_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:25_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:25_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">q1</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">q1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a>, <a class="ref" href="nat_3.html#T36">NAT_3:36</a></i>;<br/>
<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="E5:25_1"/><i><font color="Green">E55</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">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>, </i>;<br/>
<a NAME="E6:25_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E7:25_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">n1</font> = <font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:25_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="nat_4.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <font color="Maroon">d</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font>;<br/>
<a NAME="E11:25_1"/><i><font color="Green">E60</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">d</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:25_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">d</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">d</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c</font>
 
;<br/>
<a NAME="E3:25_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:25_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">q1</font> is <a href="int_2.html#V1">prime</a>
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E14:25_1"/>
not <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<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
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<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
 ;<br/>

<a NAME="E2:25_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <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>
 
<b>by </b><i>, <a class="ref" href="newton.html#T63">NEWTON:63</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:25_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="nat_4.html#E1:25"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="nat_4.html#E2:25"><i><font color="Green">E39</font></i></a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:25"/>
<font color="Maroon">p</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:25"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E11:25"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E12:25"/><i><font color="Green">E66</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> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q1</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><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="E1:25_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">q1</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1"/>
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span> or  not <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:25_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:25_2_1_1"/><b>then </b>
<font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/><a NAME="E4:25_2_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:25_2_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span></span>)</span> <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/><a NAME="E6:25_2_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">q1</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="E7:25_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="pepin.html#T26">PEPIN:26</a></i>;<br/><a NAME="E8:25_2_1_1"/><b>then </b>
<font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span> <a href="int_1.html#R2">divides</a> <font color="Maroon">p</font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i>, <a class="ref" href="pepin.html#T8">PEPIN:8</a></i>;<br/><a NAME="E9:25_2_1_1"/><b>then </b>
<font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="scpinvar.html#T2">SCPINVAR:2</a></i>;<br/><a NAME="E10:25_2_1_1"/><b>then </b>
<font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:25"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="nat_3.html#T30">NAT_3:30</a></i>;<br/><b>hence </b><a NAME="E11:25_2_1_1"/>
<font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_3.html#T25">NAT_3:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_2"/>
not <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:25_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:25_2_1_2"/>
<font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:25"/>
 ex <font color="Olive">n2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_2.html#K24">*</a> <font color="Olive">n2</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:25"/><b>then </b>
<font color="Maroon">f</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E15:25"/><b>then </b>
<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E16:25"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <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/>
<a NAME="E17:25"/><b>then </b>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E18:25"/><b>then </b>
<font color="Maroon">p</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="pepin.html#K2">^2</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>
<a NAME="E19:25"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">p</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/>
<a NAME="E20:25"/>
<font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E21:25"/><b>then </b>
<font color="Maroon">f</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E22:25"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">p</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E23:25"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E24:25"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E25:25"/><b>then </b>
<font color="Maroon">c</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <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/>
<a NAME="E26:25"/><b>then </b>
<font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E27:25"/><b>then </b>
<font color="Maroon">f</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E28:25"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <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/>
<a NAME="E29:25"/><b>then </b>
<font color="Maroon">p</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="nat_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E30:25"/><b>then </b>
<font color="Maroon">p</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E31:25"/>
contradiction
 <b>by </b><i>, <a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/>


</div>
