<?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" title="c2">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>  <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> ;<br/><b>reconsider </b><font color="Maroon" title="c3">M</font> =  <a href="gcd_1.html#K3" title="GCD_1:func.3">Classes</a> <font color="Maroon" title="c2">R</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E2:38_1_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">A</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E3:38_1_1_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D6" target="_self" title="GCD_1:def.6">Def6</a></i>;<br/>
<b>thus </b><a NAME="E4:38_1_1_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:38_1_1_1_1"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:38_1_1_1"/>
for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">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" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_1_2"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Y</font> )
 ;<br/>

<b>assume </b><a NAME="E2:38_1_1_1_2"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">A</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E4:38_1_1_1_2"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:38_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="gcd_1.html#D6" target="_self" title="GCD_1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">B</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E6:38_1_1_1_2"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:38_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="gcd_1.html#D6" target="_self" title="GCD_1:def.6">Def6</a></i>;<br/>
<b>thus </b><a NAME="E7:38_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:38_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="gcd_1.html#E2:38_1_1_1_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="gcd_1.html#E4:38_1_1_1_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="gcd_1.html#E6:38_1_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="gcd_1.html#T20" target="_self" title="GCD_1:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c4">AmpS'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:38_1_1_1"/><i><font color="Green" title="E27">A7</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c4">AmpS'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:38_1_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="wellord2.html#T27" title="WELLORD2:th.27">WELLORD2:27</a></i>;<br/><a NAME="E6:38_1_1_1"/>
not <font color="Maroon" title="c4">AmpS'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="GCD_1:func.2">Class</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D6" target="_self" title="GCD_1:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_3"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c4">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:38_1_1_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:38_1_1_1_3"/>
not <font color="Maroon" title="c4">AmpS'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:38_1_1_1_3"><i><font color="Green" title="E28">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">AmpS'</font> = <font color="Maroon" title="c4">AmpS'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>set </b><font color="Maroon" title="c6">AmpS</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span> ;<br/><a NAME="E8:38_1_1_1"/><i><font color="Green" title="E28">A9</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  ex <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b4">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  st <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b4">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="c7">X</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>;<br/>

<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:38_1_1_1_4"/><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c5">AmpS'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:38_1_1_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<a NAME="E3:38_1_1_1_4"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K3" title="GCD_1:func.3">Classes</a> <font color="Maroon" title="c2">R</font>
 
;<br/>
<a NAME="E4:38_1_1_1_4"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
<font color="Maroon" title="c7">X</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T21" target="_self" title="GCD_1:th.21">Th21</a></i>;<br/>
<a NAME="E5:38_1_1_1_4"/><i><font color="Green" title="E31">A12</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:38_1_1_1_4"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AmpS'</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:38_1_1_1_4"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:38_1_1_1_4"/><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:38_1_1_1_4"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="gcd_1.html#E5:38_1_1_1_4"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_1_4"/><i><font color="Green" title="E34">A15</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:38_1_1_1_4"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="gcd_1.html#E4:38_1_1_1_4"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="gcd_1.html#E6:38_1_1_1_4"><i><font color="Green" title="E32">A13</font></i></a></i>;<br/>
<a NAME="E9:38_1_1_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E35">A16</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" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:38_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_1_4_1_1_1"/>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E7:38_1_1_1_4"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="gcd_1.html#E8:38_1_1_1_4"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E36">A17</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font>
 ;<br/><a NAME="E2:38_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E37">A18</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:38_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:38_1_1_1_4_1_1_2"><i><font color="Green" title="E36">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">zz</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font><b> such that </b><br/><a NAME="E5:38_1_1_1_4_1_1_2"/><i><font color="Green" title="E38">A19</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">zz</font> &amp;  ex <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">zz</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ;<br/><b>thus </b><a NAME="E6:38_1_1_1_4_1_1_2"/>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:38_1_1_1_4"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="gcd_1.html#E2:38_1_1_1_4_1_1_2"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="gcd_1.html#E5:38_1_1_1_4_1_1_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> iff <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:38_1_1_1_4_1_1"><i><font color="Green" title="E35">A16</font></i></a></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" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">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" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b3">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  st <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b3">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E8:38_1_1_1_4"><i><font color="Green" title="E34">A15</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:38_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">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" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="E29">A20</font></i>: 
 <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D6" target="_self" title="GCD_1:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_5_1"/><i><font color="Green" title="E30">A21</font></i>: 
<font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:38_1_1_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<a NAME="E4:38_1_1_1_5_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:38_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AmpS'</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E3:38_1_1_1_5_1"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:38_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:38_1_1_1_5_1"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="gcd_1.html#E3:38_1_1_1_5_1"><i><font color="Green" title="E30">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:38_1_1_1_5_1"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">AmpS</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">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" title="c8">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">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" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">AmpS</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font><b> such that </b><br/><a NAME="E3:38_1_1_1_5_2_1"/><i><font color="Green" title="E29">A22</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp;  ex <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c10">X</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font>;<br/><a NAME="E5:38_1_1_1_5_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:38_1_1_1_5_2_1"/>
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">AmpS</font> implies <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:38_1_1_1_5_2_1"><i><font color="Green" title="E29">A22</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:38_1_1_1_5_2"/>
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">AmpS</font> implies <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">AmpS</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">AmpS'</font> :  ex <font color="Olive" title="b2">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">AmpS'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> ;<br/><a NAME="E11:38_1_1_1"/><i><font color="Green" title="E29">A23</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>  ex <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">AmpS</font> st <font color="Olive" title="b2">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Olive" title="b1">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" title="c8">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">N</font> =  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> <b>by </b><i><a class="ref" href="gcd_1.html#D6" target="_self" title="GCD_1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">AmpS</font><b> such that </b><br/><a NAME="E3:38_1_1_1_6"/><i><font color="Green" title="E30">A24</font></i>: 
<font color="Maroon" title="c7">AmpS</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c9">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E8:38_1_1_1"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/>
<a NAME="E4:38_1_1_1_6"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:38_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E3:38_1_1_1_6"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:38_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:38_1_1_1_6"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">AmpS</font> st <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c8">a</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E12:38_1_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">AmpS</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Olive" title="b2">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" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">AmpS</font>;<br/>

<b>assume </b><a NAME="E1:38_1_1_1_7"/><i><font color="Green" title="E30">A25</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font>
 ;<br/>

<b>assume </b><a NAME="E2:38_1_1_1_7"/><i><font color="Green" title="E31">A26</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c9">y</font>
 ;<br/>

<a NAME="E3:38_1_1_1_7"/>
<font color="Maroon" title="c8">x</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c8">x</font>
 
;<br/>
<a NAME="E4:38_1_1_1_7"/><b>then </b><i><font color="Green" title="E32">A27</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>
<a NAME="E5:38_1_1_1_7"/><i><font color="Green" title="E33">A28</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:38_1_1_1_7"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="gcd_1.html#D5" target="_self" title="GCD_1:def.5">Def5</a></i>;<br/>
<a NAME="E6:38_1_1_1_7"/><i><font color="Green" title="E34">A29</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">AmpS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E4:38_1_1_1_7"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:38_1_1_1_7"/><i><font color="Green" title="E35">A30</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">AmpS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E5:38_1_1_1_7"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_1_7"/>
 <a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D6" target="_self" title="GCD_1:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">AmpS</font><b> such that </b><br/><a NAME="E10:38_1_1_1_7"/><i><font color="Green" title="E36">A31</font></i>: 
<font color="Maroon" title="c7">AmpS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K2" title="GCD_1:func.2">Class</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E8:38_1_1_1"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/>
<a NAME="E11:38_1_1_1_7"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E7:38_1_1_1_7"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="gcd_1.html#E10:38_1_1_1_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:38_1_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:38_1_1_1_7"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="gcd_1.html#E6:38_1_1_1_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="gcd_1.html#E10:38_1_1_1_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="b1">s</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font> st <br/>( ( for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">s</font> st <font color="Olive" title="b3">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Olive" title="b2">a</font> ) &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">s</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> holds <br/><font color="Olive" title="b2">x</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Olive" title="b3">y</font> ) )
 <b>by </b><i><a class="txt" href="gcd_1.html#E11:38_1_1_1"><i><font color="Green" title="E29">A23</font></i></a></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" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <br/>( ( for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Olive" title="b3">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Olive" title="b2">a</font> ) &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive">b<sub>1</sub></font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> holds <br/><font color="Olive" title="b2">x</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Olive" title="b3">y</font> ) )
 ;<br/>


</div>
