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

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : not <font color="Olive">a</font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:52"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : not <font color="Olive">a</font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:52_1"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : not <font color="Olive">a</font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> 
 ;<br/>

<a NAME="E3:52_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:52_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49">GROUP_2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E6:52_1"/>
<font color="Maroon">a</font> is <a href="group_4.html#V1" target="_self">generating</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:52_1"/><i><font color="Green">E34</font></i>: 
 <a href="group_4.html#K5" target="_self">gr</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E9:52_1"/><i><font color="Green">E37</font></i>: 
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">M</font> = <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>] <b>means </b>( <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:52_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E12:52_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> holds <br/> ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] )
 <b>from </b><i><a class="ref" href="group_4.html#S2" target="_self">GROUP_4:sch 2</a>();<br/></i>
<a NAME="E13:52_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:52_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><a NAME="E1:52_1_1_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="group_4.html#E9:52_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>, <font color="Maroon">I</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:52_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:52_1_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:52_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:52_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font></span>)</span>
 <b>and </b><br/><a NAME="E4:52_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E6:52_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:52_1_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="group_4.html#E5:52_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:52_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_4.html#K2" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_4.html#E3:52_1_1_1_1"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_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="E1:52_1_1_1_1_1_1"/>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, <a class="txt" href="group_4.html#E7:52_1_1_1_1"><i><font color="Green">E48</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:52_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_1_1_2"/>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:52_1_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="txt" href="group_4.html#E8:52_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E7:52_1_1_1_1"><i><font color="Green">E48</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:52_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1: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="E10:52_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:52_1_1_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_1_1"><i><font color="Green">E44</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E11:52_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="group_2.html#T71">GROUP_2:71</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Olive">H2</font> );<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:52_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E2:52_1_2"/>
<font color="Maroon">x</font> is  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:52_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_3"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">x</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E2:52_1_3"/>
<font color="Maroon">H</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T63">GROUP_2:63</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:52_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T64">GROUP_2:64</a></i>;<br/>
<a NAME="E20:52_1"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_4"><b>proof </b></a><div class="add">

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

<b>given </b><font color="Maroon">H1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>, <font color="Maroon">H2</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E1:52_1_4"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">H1</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">H2</font> )
 ;<br/>

<b>given </b><font color="Maroon">H3</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>, <font color="Maroon">H4</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:52_1_4"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H3</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H4</font> &amp; <font color="Maroon">H3</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">H4</font> )
 ;<br/>

<a NAME="E3:52_1_4"/>
<font color="Maroon">H1</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">H4</font>
 
<b>by </b><i>, , <a class="ref" href="group_2.html#T65">GROUP_2:65</a></i>;<br/>
<b>hence </b><a NAME="E4:52_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:52_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) holds <br/> ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_1_5"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><a NAME="E2:52_1_5"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 ;<br/>

<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_5_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> 
 ;<br/><a NAME="E2:52_1_5_1"/><b>then </b>
 ex <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> &amp;  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) )
 ;<br/><b>hence </b><a NAME="E3:52_1_5_1"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">E</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_5_2_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E3:52_1_5_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>hence </b><a NAME="E4:52_1_5_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><a NAME="E3:52_1_5_2_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/><a NAME="E5:52_1_5_2_1_2"/><i><font color="Green">E120</font></i>: 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:52_1_5_2_1_2"/><b>then </b>
(  <a href="group_2.html#K7">carr</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> &amp; <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:52_1_5_2_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:52_1_5_2_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E17:52_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="orders_1.html#T91">ORDERS_1:91</a></i>;<br/><div><i><font color="Green">E123</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:52_1_5_2_1_2_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>and </b><br/><a NAME="E2:52_1_5_2_1_2_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/><b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:52_1_5_2_1_2_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:52_1_5_2_1_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:52_1_5_2_1_2_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E8:52_1_5_2_1_2_1"/><i><font color="Green">E129</font></i>: 
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">H1</font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E10:52_1_5_2_1_2_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:52_1_5_2_1_2_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E13:52_1_5_2_1_2_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E15:52_1_5_2_1_2_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E16:52_1_5_2_1_2_1"/><i><font color="Green">E134</font></i>: 
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">H2</font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E18:52_1_5_2_1_2_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_5_2_1_2_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H1</font>,<font color="Maroon">H2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H2</font>,<font color="Maroon">H1</font>] )
 </span><b>by </b><i>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H1</font>,<font color="Maroon">H2</font>]
 ;<br/><div class="add"><a NAME="E2:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><a NAME="E3:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E4:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font>
 ;<br/><a NAME="E5:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font>
 <b>by </b><i>, , , , , , <a class="ref" href="group_2.html#T89">GROUP_2:89</a></i>;<br/><b>hence </b><a NAME="E6:52_1_5_2_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H2</font>,<font color="Maroon">H1</font>]
 ;<br/><div class="add"><a NAME="E2:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><a NAME="E3:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E4:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font>
 ;<br/><a NAME="E5:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font>
 <b>by </b><i>, , , , , , <a class="ref" href="group_2.html#T89">GROUP_2:89</a></i>;<br/><b>hence </b><a NAME="E6:52_1_5_2_1_2_1_1_1_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:52_1_5_2_1_2_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:52_1_5_2_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_5_2_1_2_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:52_1_5_2_1_2_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:52_1_5_2_1_2_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E7:52_1_5_2_1_2_2"/><i><font color="Green">E139</font></i>: 
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">H1</font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:52_1_5_2_1_2_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i/>;<br/><a NAME="E10:52_1_5_2_1_2_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font>
 <b>by </b><i>, , , <a class="ref" href="group_2.html#T90">GROUP_2:90</a></i>;<br/><b>hence </b><a NAME="E11:52_1_5_2_1_2_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">H</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E12:52_1_5_2_1_2"/><i><font color="Green">E141</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="txt" href="group_4.html#E19:52_1"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/><div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_5_2_1_2_3"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:52_1_5_2_1_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:52_1_5_2_1_2_3"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:52_1_5_2_1_2_3"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:52_1_5_2_1_2_3"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E8:52_1_5_2_1_2_3"/><i><font color="Green">E163</font></i>: 
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">H1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E10:52_1_5_2_1_2_3"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i/>;<br/><a NAME="E11:52_1_5_2_1_2_3"/>
not <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 <b>by </b><i><a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E12:52_1_5_2_1_2_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E14:52_1_5_2_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">H</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E18:52_1"><i><font color="Green">E52</font></i></a>, , </i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E16:52_1_5_2_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E17:52_1_5_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">x</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/><b>take </b><font color="Maroon">K</font> = <font color="Maroon">K</font>;<br/><b>take </b><font color="Maroon">H</font> = <font color="Maroon">H</font>;<br/><b>thus </b><a NAME="E19:52_1_5_2_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> )
 ;<br/><a NAME="E20:52_1_5_2_1_2"/>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 ;<br/><a NAME="E21:52_1_5_2_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> :  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E22:52_1_5_2_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E23:52_1_5_2_1_2"/>
<font color="Maroon">K</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T66">GROUP_2:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:52_1_5"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E23:52_1"/><i><font color="Green">E166</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/> not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">s</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1">ORDERS_1:sch 1</a>(, , , );<br/></i>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">s</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E25:52_1"/>
<font color="Maroon">H</font> is <a href="group_4.html#V2" target="_self">maximal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:52_1_6"/>
not <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:52_1_6"/>
 <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> #) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #)
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_4.html#D7" target="_self">GROUP_4:def 7</a><br/>

<b>let </b><font color="Maroon">K</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:52_1_6"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> #)
 <b>and </b><br/><a NAME="E4:52_1_6"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">H</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E5:52_1_6"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #)
 ;<br/>

<a NAME="E6:52_1_6"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">K</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> &amp; <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E7:52_1_6"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_6_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:52_1_6_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:52_1_6_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:52_1_6_1"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="group_4.html#E17:52_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E5:52_1_6_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_6_1_1"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_6_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="E1:52_1_6_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_6_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_6_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:52_1_6_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_6_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_6_1_1_1_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:52_1_6_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="group_4.html#E19:52_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:52_1_6_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_1_6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:52_1_6_1"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="group_4.html#E18:52_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:52_1_6_1"/><b>then </b>
 <a href="group_4.html#K5" target="_self">gr</a> <font color="Maroon">B</font> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">K</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E8:52_1_6_1"/><b>then </b>
<font color="Maroon">G</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="group_4.html#E9:52_1"><i><font color="Green">E37</font></i></a>, </i>;<br/><b>hence </b><a NAME="E9:52_1_6_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="group_2.html#T64">GROUP_2:64</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">K</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E17:52_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E10:52_1_6"/>
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">v</font>]
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:52_1_6"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E26:52_1"/><b>then </b>
 <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="txt" href="group_4.html#E2:52_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E27:52_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E28:52_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 
<b>by </b><i>, <a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E29:52_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:52"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : not <font color="Olive">a</font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:52"/><i><font color="Green">E175</font></i>: 
not <font color="Maroon">a</font> is <a href="group_4.html#V1" target="_self">generating</a>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_2_1"/>
( for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  holds not <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> or  ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1"><b>suppose </b></a><a NAME="E1:52_2_1_1"/>
for <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  holds not <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a>
 ;<br/><div class="add"><a NAME="E2:52_2_1_1"/><b>then </b>
 <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:52_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_2"><b>suppose </b></a><a NAME="E1:52_2_1_2"/><i><font color="Green">E176</font></i>: 
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <font color="Olive">H</font> is <a href="group_4.html#V2" target="_self">maximal</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:52_2_1_2_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">H</font> is <a href="group_4.html#V2" target="_self">maximal</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2_1_2_1_1"/><i><font color="Green">E178</font></i>: 
not <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:52_2_1_2_1_1"/><b>then </b><i><font color="Green">E179</font></i>: 
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:52_2_1_2_1_1"/><i><font color="Green">E180</font></i>: 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_2_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:52_2_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:52_2_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:52_2_1_2_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> = 
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
;<br/><a NAME="E5:52_2_1_2_1_1"/><b>then </b>
(  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="group_2.html#R1">=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:52_2_1_2_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_2_1_2_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:52_2_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:52_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1: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="E7:52"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>


</div>
