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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:73"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:73"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<a NAME="E3:73"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E4:73"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E5:73"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="txt" href="wsierp_1.html#E3:73"><i><font color="Green">E53</font></i></a>, <a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E6:73"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E7:73"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="txt" href="wsierp_1.html#E4:73"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wsierp_1.html#T37" target="_self">Th37</a></i>;<br/>
<b>then 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="E9:73"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:73_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E3:73_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E4:73_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E5:73_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> 0 )
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E6:73_1"/>
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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:73_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<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="E4:73_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:73_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="int_2.html#T5">INT_2:5</a></i>;<br/><a NAME="E3:73_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><a NAME="E4:73_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>;<br/><a NAME="E5:73_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:73_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="wsierp_1.html#E2:73_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E6:73_2_1"/>
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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:73_2_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_2_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:73_2_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E3:73_2_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_2_2_1"/><i><font color="Green">E64</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:73_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="txt" href="wsierp_1.html#E2:73_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><a NAME="E3:73_2_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:73_2_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="wsierp_1.html#E1:73_2_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:73_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><a NAME="E5:73_2_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>;<br/><a NAME="E6:73_2_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:73_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="wsierp_1.html#E3:73_2_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E7:73_2_2_1"/>
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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:73_2_2_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_2_2_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:73_2_2_2"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:73_2_2_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="int_2.html#R1">are_relative_prime</a>  )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:73"><i><font color="Green">E52</font></i></a>, <a class="txt" href="wsierp_1.html#E1:73_2_2_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="int_2.html#T38">INT_2:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="wsierp_1.html#E3:73_2_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="wsierp_1.html#E1:73_2_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="wsierp_1.html#E4:73_2_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E6:73_2_2_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:73_2_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="wsierp_1.html#E4:73_2_2_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E7:73_2_2_2"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:73_2_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="wsierp_1.html#E19"><i><font color="Green">Lemma14</font></i></a></i>;<br/><a NAME="E8:73_2_2_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E9:73"><i><font color="Green">E55</font></i></a>, <a class="txt" href="wsierp_1.html#E4:73_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="wsierp_1.html#E1:73_2_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="wsierp_1.html#E4:73_2_2_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</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="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</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="wsierp_1.html#E1:73_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
;<br/><a NAME="E10:73_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:73_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8:73_2_2_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E12:73_2_2_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma22</font></i></a></i>;<br/><i><font color="Green">E70</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E6:73_2_2_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="prepower.html#T15">PREPOWER:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:73"><i><font color="Green">E51</font></i></a>, <a class="txt" href="wsierp_1.html#E9:73"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:73_2_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
;<br/><a NAME="E14:73_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>11</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E15:73_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12:73_2_2_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="newton.html#T62">NEWTON:62</a></i>;<br/><b>hence </b><a NAME="E16:73_2_2_2"/>
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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9:73"><i><font color="Green">E55</font></i></a>, <a class="txt" href="wsierp_1.html#E13:73_2_2_2"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:73_2_2"/>
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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:73_2_2"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:73_2"/>
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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:73_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:73"/>
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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E10:73"><i><font color="Green">E56</font></i></a></i>;<br/>


</div>
