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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">A</font> be    <a href="gcd_1.html#M1" title="GCD_1:mode.1">Am</a> of <font color="Maroon" title="c1">R</font>;<br/><b>consider </b><font color="Maroon" title="c3">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E2:39_1_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D7" target="_self" title="GCD_1:def.7">Def7</a></i>;<br/><b>set </b><font color="Maroon" title="c4">A'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E3:39_1_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_1_1"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></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="E2:39_1_1_1_1"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c5">A'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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/><a NAME="E5:39_1_1_1"/><i><font color="Green" title="E29">A3</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="c5">A'</font>  holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">A'</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_2_1_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span>  or <font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">zz</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">zz</font> &amp; <font color="Olive" title="b1">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><b>hence </b><a NAME="E3:39_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_1_2"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E6:39_1_1_1"/>
<font color="Maroon" title="c5">A'</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="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1_1_3_1_1"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A'</font>
 ;<br/><a NAME="E2:39_1_1_1_3_1_1"/>
<font color="Maroon" title="c6">x</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="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 </span><b>by </b><i><a class="txt" href="gcd_1.html#E5:39_1_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="gcd_1.html#E1:39_1_1_1_3_1_1"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</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="c1">R</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">x</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="c1">R</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c6">x</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="c1">R</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:39_1_1_1_3_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A'</font> implies <font color="Maroon" title="c6">x</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="c1">R</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:39_1_1_1_3_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A'</font> implies <font color="Maroon" title="c6">x</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="c1">R</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_1_3"/>
<font color="Maroon" title="c5">A'</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="c1">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="c6">A'</font> = <font color="Maroon" title="c5">A'</font> 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="c1">R</font> ;<br/><a NAME="E8:39_1_1_1"/><i><font color="Green" title="E30">A5</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="c1">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="c6">A'</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="39_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_4_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c7">a</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_4_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_4_1_1_1"/>
 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="c6">A'</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="c7">a</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:39_1_1_1"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_4_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_4_1_1_2"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:39_1_1_1_4_1_1_2"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D7" target="_self" title="GCD_1:def.7">Def7</a></i>;<br/><a NAME="E4:39_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:39_1_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="gcd_1.html#E1:39_1_1_1_4_1_1_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="gcd_1.html#E3:39_1_1_1_4_1_1_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="gcd_1.html#T4" target="_self" title="GCD_1:th.4">Th4</a></i>;<br/><a NAME="E5:39_1_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">zz</font> where <font color="Olive" title="b1">zz</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span> 
 ;<br/><a NAME="E6:39_1_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_1_1_4_1_1_2"/>
 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="c6">A'</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="c7">a</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:39_1_1_1_4_1_1_2"><i><font color="Green" title="E32">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_1_4"/>
 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="c6">A'</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="c7">a</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E9:39_1_1_1"/>
for <font color="Olive" title="b1">z</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="c6">A'</font>  st <font color="Olive" title="b1">z</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">z</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="39_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">z</font>, <font color="Maroon" title="c8">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">A'</font>;<br/>

<b>assume </b><a NAME="E1:39_1_1_1_5"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_5_1_1"/>
( ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_5_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1_5_1_1_1"/>
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_5_1_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:39_1_1_1_5"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_5_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_1_5_1_1_2"/><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_5_1_1_2"/><b>then </b>
not <font color="Maroon" title="c8">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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></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="E3:39_1_1_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">zz</font> where <font color="Olive" title="b1">zz</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:39_1_1_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
 ex <font color="Olive" title="b1">zz</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">zz</font> &amp; <font color="Olive" title="b1">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><b>assume </b><a NAME="E5:39_1_1_1_5_1_1_2"/>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c8">y</font>
 ;<br/><a NAME="E6:39_1_1_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:39_1_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="gcd_1.html#E1:39_1_1_1_5_1_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="gcd_1.html#T4" target="_self" title="GCD_1:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E7:39_1_1_1_5_1_1_2"/>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E4:39_1_1_1_5_1_1_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="gcd_1.html#D7" target="_self" title="GCD_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_5_1_1_3"><b>case </b></a><a NAME="E1:39_1_1_1_5_1_1_3"/><i><font color="Green" title="E32">A11</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_5_1_1_3"/><b>then </b>
not <font color="Maroon" title="c7">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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></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="E3:39_1_1_1_5_1_1_3"/><b>then </b>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">zz</font> where <font color="Olive" title="b1">zz</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> : <font color="Olive" title="b1">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:39_1_1_1_5_1_1_3"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
 ex <font color="Olive" title="b1">zz</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">zz</font> &amp; <font color="Olive" title="b1">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><b>assume </b><a NAME="E5:39_1_1_1_5_1_1_3"/>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c8">y</font>
 ;<br/><a NAME="E6:39_1_1_1_5_1_1_3"/><b>then </b>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:39_1_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="gcd_1.html#E1:39_1_1_1_5_1_1_3"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="gcd_1.html#T4" target="_self" title="GCD_1:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E7:39_1_1_1_5_1_1_3"/>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E4:39_1_1_1_5_1_1_3"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="gcd_1.html#D7" target="_self" title="GCD_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_5_1_1_4"><b>case </b></a><a NAME="E1:39_1_1_1_5_1_1_4"/><i><font color="Green" title="E32">A13</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_5_1_1_4"/><b>then </b><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:39_1_1_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/><a NAME="E3:39_1_1_1_5_1_1_4"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:39_1_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="gcd_1.html#E1:39_1_1_1_5_1_1_4"><i><font color="Green" title="E32">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_1_1_1_5_1_1_4"/>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:39_1_1_1_5"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="gcd_1.html#E2:39_1_1_1_5_1_1_4"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="gcd_1.html#D7" target="_self" title="GCD_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_1_1_1_5"/>
<font color="Maroon" title="c7">z</font> <a href="gcd_1.html#NR4" title="GCD_1:NR.4">is_not_associated_to</a> <font color="Maroon" title="c8">y</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E10:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">A'</font> is    <a href="gcd_1.html#M1" title="GCD_1:mode.1">Am</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E8:39_1_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="gcd_1.html#D7" target="_self" title="GCD_1:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E11:39_1_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/>( <font color="Olive">b<sub>1</sub></font> is    <a href="gcd_1.html#M1" title="GCD_1:mode.1">Am</a> of <font color="Maroon" title="c1">R</font> &amp;  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:39_1_1_1"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_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/>( <font color="Olive">b<sub>1</sub></font> is    <a href="gcd_1.html#M1" title="GCD_1:mode.1">Am</a> of <font color="Maroon" title="c1">R</font> &amp;  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


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