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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a>  <a href="vectsp_1.html#L1">multLoopStr</a> ;<br/><b>reconsider </b><font color="Maroon">M</font> =  <a href="gcd_1.html#K3" target="_self">Classes</a> <font color="Maroon">R</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E2:38_1_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:38_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:38_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E3:38_1_1_1"/>
for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> holds <br/><font color="Olive">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>assume </b><a NAME="E2:38_1_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:38_1_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:38_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E7:38_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">AmpS'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_1_1_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">AmpS'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="wellord2.html#T27">WELLORD2:27</a></i>;<br/><a NAME="E6:38_1_1_1"/>
not <font color="Maroon">AmpS'</font> is <a href="xboole_0.html#V1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:38_1_1_1_3"/>
 <a href="gcd_1.html#K2" target="_self">Class</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" target="_self">Class</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:38_1_1_1_3"/>
not <font color="Maroon">AmpS'</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">AmpS'</font> = <font color="Maroon">AmpS'</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>set </b><font color="Maroon">AmpS</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> ;<br/><a NAME="E8:38_1_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">X</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  st <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font>;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:38_1_1_1_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">AmpS'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E3:38_1_1_1_4"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="gcd_1.html#K3" target="_self">Classes</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E4:38_1_1_1_4"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">X</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:38_1_1_1_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:38_1_1_1_4"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AmpS'</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:38_1_1_1_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="gcd_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_1_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:38_1_1_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_1_4_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:38_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_1_4_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="gcd_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_4_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_1_4_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:38_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:38_1_1_1_4_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">zz</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font><b> such that </b><br/><a NAME="E5:38_1_1_1_4_1_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">zz</font> &amp;  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">zz</font></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><b>thus </b><a NAME="E6:38_1_1_1_4_1_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="gcd_1.html#D3" target="_self">Def3</a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_4_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:38_1_1_1_4_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:38_1_1_1_4"/>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  st <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:38_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:38_1_1_1_5"/>
not <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:38_1_1_1_5_1"/><i><font color="Green">E93</font></i>: 
 <a href="gcd_1.html#K2" target="_self">Class</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_5_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" target="_self">Class</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E4:38_1_1_1_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:38_1_1_1_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AmpS'</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:38_1_1_1_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:38_1_1_1_5_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">AmpS</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_5_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_1_5_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AmpS</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font><b> such that </b><br/><a NAME="E3:38_1_1_1_5_2_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon">X</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:38_1_1_1_5_2_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i/>;<br/><a NAME="E6:38_1_1_1_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:38_1_1_1_5_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E8:38_1_1_1_5_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AmpS</font> implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:38_1_1_1_5_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AmpS</font> implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38_1_1_1_5"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">AmpS</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AmpS'</font> :  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">AmpS'</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/><a NAME="E11:38_1_1_1"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">AmpS</font> st <font color="Olive">z</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>reconsider </b><font color="Maroon">N</font> =  <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">AmpS</font><b> such that </b><br/><a NAME="E3:38_1_1_1_6"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">AmpS</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E4:38_1_1_1_6"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:38_1_1_1_6"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:38_1_1_1_6"/><b>then </b>
<font color="Maroon">z</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:38_1_1_1_6"/>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">AmpS</font> st <font color="Olive">z</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E12:38_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">AmpS</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="gcd_1.html#NR4" target="_self">is_not_associated_to</a> <font color="Olive">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">AmpS</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">AmpS</font>;<br/>

<b>assume </b><a NAME="E1:38_1_1_1_7"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>assume </b><a NAME="E2:38_1_1_1_7"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:38_1_1_1_7"/>
<font color="Maroon">x</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E4:38_1_1_1_7"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:38_1_1_1_7"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E6:38_1_1_1_7"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AmpS</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:38_1_1_1_7"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AmpS</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_1_7"/>
 <a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">AmpS</font><b> such that </b><br/><a NAME="E10:38_1_1_1_7"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">AmpS</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" target="_self">Class</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E11:38_1_1_1_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:38_1_1_1_7"/>
contradiction
 <b>by </b><i>, <a class="ref" href="gcd_1.html#D1" target="_self">Def1</a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:38_1_1_1"/>
 ex <font color="Olive">s</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Olive">s</font> st <font color="Olive">z</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Olive">a</font> ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Olive">s</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="gcd_1.html#NR4" target="_self">is_not_associated_to</a> <font color="Olive">y</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:38_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Olive">z</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Olive">a</font> ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="gcd_1.html#NR4" target="_self">is_not_associated_to</a> <font color="Olive">y</font> ) )
 ;<br/>


</div>
