<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:108"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E2:108"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>2</sub></font> ) );<br/>
<a NAME="E3:108"/><i><font color="Green">E64</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="newton.html#E1:108"><i><font color="Green">E62</font></i></a>, <a class="txt" href="newton.html#E2:108"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:108"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>and </b><br/><a NAME="E6:108"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="newton.html#E3:108"><i><font color="Green">E64</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E7:108"/><i><font color="Green">E67</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="newton.html#E5:108"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:108"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 <b>and </b><br/><a NAME="E10:108"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="newton.html#E7:108"><i><font color="Green">E67</font></i></a>);<br/></i>
<a NAME="E11:108"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E9:108"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E12:108"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:108"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="newton.html#E9:108"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E15:108"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:108_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E5:108"><i><font color="Green">E65</font></i></a>, <a class="txt" href="newton.html#E107"><i><font color="Green">Lemma61</font></i></a></i>;<br/>
<a NAME="E3:108_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E4:108_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E1:108_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:108_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E6:108_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E2:108_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E7:108_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:108_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E2:108_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E2:108_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E3:108_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:108_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="newton.html#E14:108"><i><font color="Green">E71</font></i></a>, <a class="txt" href="newton.html#E1:108_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T57">NAT_1:57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:108_1"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E14:108"><i><font color="Green">E71</font></i></a>, <a class="txt" href="newton.html#E5:108_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="newton.html#E6:108_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="nat_1.html#T55">NAT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E9:108_1"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E10:108"><i><font color="Green">E69</font></i></a>, <a class="txt" href="newton.html#E14:108"><i><font color="Green">E71</font></i></a>, <a class="txt" href="newton.html#E4:108_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="newton.html#E7:108_1"><i><font color="Green">E78</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:108"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:108_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/>
<a NAME="E4:108_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:108_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:108_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E12:108"><i><font color="Green">E70</font></i></a>, <a class="txt" href="newton.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E7:108_2"/><b>then </b><i><font color="Green">E77</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E15:108"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:108_2"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:108_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="newton.html#E5:108"><i><font color="Green">E65</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:108_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E12:108"><i><font color="Green">E70</font></i></a>, <a class="txt" href="newton.html#E15:108"><i><font color="Green">E72</font></i></a>, <a class="txt" href="newton.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:108_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="newton.html#E14:108"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E11:108_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="newton.html#E10:108_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<a NAME="E12:108_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E6:108"><i><font color="Green">E66</font></i></a>, <a class="txt" href="newton.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="newton.html#E7:108_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="newton.html#E8:108_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:108_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:108_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="newton.html#E3:108_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E17:108_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:108_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E9:108"><i><font color="Green">E68</font></i></a>, <a class="txt" href="newton.html#E16:108_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:108_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:108_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:108_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E16:108_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E2:108_2_3"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E12:108"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E3:108_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_2_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:108_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:108_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E4:108_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="newton.html#E16:108_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="newton.html#E17:108_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:108_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="newton.html#E2:108_2_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:108_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:108_2_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E16:108_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E2:108_2_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E10:108_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E3:108_2_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E14:108_2"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E4:108_2_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
;<br/>
<a NAME="E5:108_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
;<br/>
<a NAME="E6:108_2_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E5:108_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E7:108_2_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:108_2"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E14:108"><i><font color="Green">E71</font></i></a>, <a class="txt" href="newton.html#E16:108_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<b>hence </b><a NAME="E21:108_2"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E10:108"><i><font color="Green">E69</font></i></a>, <a class="txt" href="newton.html#E14:108"><i><font color="Green">E71</font></i></a>, <a class="txt" href="newton.html#E18:108_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="newton.html#E19:108_2"><i><font color="Green">E84</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:108"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/>

<a NAME="E2:108_3"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
;<br/>
<a NAME="E3:108_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E12:108"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:108_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E2:108_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<b>hence </b><a NAME="E5:108_3"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E14:108"><i><font color="Green">E71</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:108"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E16:108"><i><font color="Green">E73</font></i></a></i>;<br/>


</div>
