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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>

<b>set </b><font color="Maroon" title="c2">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : not <font color="Olive" title="b1">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:52"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : not <font color="Olive" title="b1">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:52_1"/><i><font color="Green" title="E34">A2</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : not <font color="Olive" title="b1">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> </span>}</span> 
 ;<br/>

<a NAME="E3:52_1"/>
<font color="Maroon" title="c3">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_4.html#E1:52_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E4:52_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49" title="GROUP_2:th.49">GROUP_2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:52_1"/>
<font color="Maroon" title="c4">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a>
 
<b>by </b><i><a class="txt" href="group_4.html#E2:52_1"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E8:52_1"/><i><font color="Green" title="E35">A3</font></i>: 
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E9:52_1"/><i><font color="Green" title="E36">A4</font></i>: 
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_4.html#D6" target="_self" title="GROUP_4:def.6">Def6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>] <b>means </b>( <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> $1 &amp; not <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> $1 );<br/>
<b>consider </b><font color="Maroon" title="c7">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:52_1"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E12:52_1"/><i><font color="Green" title="E38">A6</font></i>: 
for <font color="Olive" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] )
 <b>from </b><i><a class="ref" href="group_4.html#S2" target="_self" title="GROUP_4:sch.2">GROUP_4:sch 2</a>();<br/></i>
<a NAME="E13:52_1"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#D5" target="_self" title="GROUP_4:def.5">Def5</a></i>;<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" title="E40">A8</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><a NAME="E1:52_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c10">I</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E3:52_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F</font>
 <b>and </b><br/><a NAME="E4:52_1_1_1"/><i><font color="Green" title="E41">A9</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E5:52_1_1_1"/><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T37" target="_self" title="GROUP_4:th.37">Th37</a></i>;<br/><a NAME="E6:52_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_1_1_1_1"/><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span>
 <b>and </b><br/><a NAME="E4:52_1_1_1_1"/><i><font color="Green" title="E44">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_4.html#E3:52_1_1_1_1"><i><font color="Green" title="E43">A11</font></i></a></i>;<br/>
<a NAME="E6:52_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F</font>
 
<b>by </b><i><a class="ref" href="group_4.html#D4" target="_self" title="GROUP_4:def.4">Def4</a></i>;<br/>
<a NAME="E7:52_1_1_1_1"/><b>then </b><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F</font>
 
<b>by </b><i><a class="txt" href="group_4.html#E3:52_1_1_1_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:52_1_1_1_1"/><b>then </b><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_4.html#E4:52_1_1_1_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="group_4.html#D4" target="_self" title="GROUP_4:def.4">Def4</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" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="group_4.html#E8:52_1_1_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="group_4.html#T6" target="_self" title="GROUP_4:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E3:52_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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" title="E47">A15</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">F</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E7:52_1_1_1_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:52_1_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_1_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="group_4.html#E1:52_1_1_1_1_1_1_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T38" target="_self" title="GROUP_4:th.38">Th38</a></i>;<br/><a NAME="E6:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_1_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="group_4.html#T6" target="_self" title="GROUP_4:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E7:52_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="c8">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_1_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="group_4.html#T21" target="_self" title="GROUP_4:th.21">Th21</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#E9:52_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="group_2.html#T71" title="GROUP_2:th.71">GROUP_2:71</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c7">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="group_4.html#E13:52_1"><i><font color="Green" title="E39">A7</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b2">H2</font> );<br/>
<div><i><font color="Green" title="E40">A16</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" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>;<br/><a NAME="E1:52_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="group_4.html#E11:52_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E2:52_1_2"/>
<font color="Maroon" title="c9">x</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:52_1"/><i><font color="Green" title="E41">A17</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="c8">X</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">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" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c9">x</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_4.html#E16:52_1"><i><font color="Green" title="E40">A16</font></i></a></i>;<br/>
<a NAME="E2:52_1_3"/>
<font color="Maroon" title="c10">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c10">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T63" title="GROUP_2:th.63">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" title="c9">x</font>,<font color="Maroon" title="c9">x</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:52_1"/><i><font color="Green" title="E42">A18</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T64" title="GROUP_2:th.64">GROUP_2:64</a></i>;<br/>
<a NAME="E19:52_1"/><i><font color="Green" title="E43">A19</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">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" title="c9">x</font>, <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c11">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>;<br/>

<b>given </b><font color="Maroon" title="c12">H1</font>, <font color="Maroon" title="c13">H2</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E2:52_1_4"/><i><font color="Green" title="E44">A20</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">H1</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">H2</font> &amp; <font color="Maroon" title="c12">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c13">H2</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c14">H3</font>, <font color="Maroon" title="c15">H4</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E4:52_1_4"/><i><font color="Green" title="E45">A21</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H3</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">H4</font> &amp; <font color="Maroon" title="c14">H3</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c15">H4</font> )
 ;<br/>

<a NAME="E5:52_1_4"/>
<font color="Maroon" title="c12">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c15">H4</font>
 
<b>by </b><i><a class="txt" href="group_4.html#E2:52_1_4"><i><font color="Green" title="E44">A20</font></i></a>, <a class="txt" href="group_4.html#E4:52_1_4"><i><font color="Green" title="E45">A21</font></i></a>, <a class="ref" href="group_2.html#T65" title="GROUP_2:th.65">GROUP_2:65</a></i>;<br/>
<b>hence </b><a NAME="E6:52_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</font>]
 <b>by </b><i><a class="txt" href="group_4.html#E2:52_1_4"><i><font color="Green" title="E44">A20</font></i></a>, <a class="txt" href="group_4.html#E4:52_1_4"><i><font color="Green" title="E45">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:52_1"/><i><font color="Green" title="E44">A22</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font> &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>] ) holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font> st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">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" title="c9">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_1_5"/><i><font color="Green" title="E45">A23</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<b>assume </b><a NAME="E2:52_1_5"/><i><font color="Green" title="E46">A24</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c10">C</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">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" title="c11">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_5_1"/>
<font color="Maroon" title="c11">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 ;<br/><a NAME="E2:52_1_5_1"/><b>then </b>
 ex <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c11">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> &amp;  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) )
 ;<br/><b>hence </b><a NAME="E3:52_1_5_1"/>
<font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">E</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T94" title="ZFMISC_1:th.94">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" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c9">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </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" title="E47">A25</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>;<br/><b>take </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">y</font>;<br/><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>;<br/><b>assume </b><a NAME="E3:52_1_5_2_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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" title="c14">x</font>,<font color="Maroon" title="c13">y</font>]
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5_2_1_1"><i><font color="Green" title="E47">A25</font></i></a></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" title="E47">A26</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c12">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Y</font>;<br/><a NAME="E3:52_1_5_2_1_2"/><i><font color="Green" title="E48">A27</font></i>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="group_4.html#E1:52_1_5_2_1_2"><i><font color="Green" title="E47">A26</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c12">s</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_4.html#E11:52_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/><a NAME="E5:52_1_5_2_1_2"/><i><font color="Green" title="E49">A28</font></i>: 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5_2_1_2"><i><font color="Green" title="E47">A26</font></i></a></i>;<br/><a NAME="E6:52_1_5_2_1_2"/><b>then </b>
(  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c13">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c13">s</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="group_4.html#E3:52_1_5_2_1_2"><i><font color="Green" title="E48">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:52_1_5_2_1_2"/><b>then </b><i><font color="Green" title="E50">A29</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:52_1_5_2_1_2"/><i><font color="Green" title="E51">A30</font></i>: 
<font color="Maroon" title="c11">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2"><i><font color="Green" title="E49">A28</font></i></a>, <a class="ref" href="orders_1.html#T91" title="ORDERS_1:th.91">ORDERS_1:91</a></i>;<br/><div><i><font color="Green" title="E52">A31</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" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:52_1_5_2_1_2_1"/><i><font color="Green" title="E53">A32</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 <b>and </b><br/><a NAME="E2:52_1_5_2_1_2_1"/><i><font color="Green" title="E54">A33</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 ;<br/><b>consider </b><font color="Maroon" title="c16">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:52_1_5_2_1_2_1"/><i><font color="Green" title="E55">A34</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font>
 <b>and </b><br/><a NAME="E5:52_1_5_2_1_2_1"/><i><font color="Green" title="E56">A35</font></i>: 
<font color="Maroon" title="c16">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5_2_1_2_1"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:52_1_5_2_1_2_1"/><i><font color="Green" title="E57">A36</font></i>: 
<font color="Maroon" title="c16">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">D</font>
 <b>and </b><br/><a NAME="E8:52_1_5_2_1_2_1"/><i><font color="Green" title="E58">A37</font></i>: 
 ex <font color="Olive" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c17">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b1">H</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2_1"><i><font color="Green" title="E56">A35</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">H1</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E10:52_1_5_2_1_2_1"/><i><font color="Green" title="E59">A38</font></i>: 
( <font color="Maroon" title="c18">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c17">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c18">H1</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_5_2_1_2_1"><i><font color="Green" title="E58">A37</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:52_1_5_2_1_2_1"/><i><font color="Green" title="E60">A39</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Z1</font>
 <b>and </b><br/><a NAME="E13:52_1_5_2_1_2_1"/><i><font color="Green" title="E61">A40</font></i>: 
<font color="Maroon" title="c19">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E2:52_1_5_2_1_2_1"><i><font color="Green" title="E54">A33</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E15:52_1_5_2_1_2_1"/><i><font color="Green" title="E62">A41</font></i>: 
<font color="Maroon" title="c19">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">B</font>
 <b>and </b><br/><a NAME="E16:52_1_5_2_1_2_1"/><i><font color="Green" title="E63">A42</font></i>: 
 ex <font color="Olive" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c20">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b1">H</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E13:52_1_5_2_1_2_1"><i><font color="Green" title="E61">A40</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">H2</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E18:52_1_5_2_1_2_1"/><i><font color="Green" title="E64">A43</font></i>: 
( <font color="Maroon" title="c21">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c20">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c21">H2</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E16:52_1_5_2_1_2_1"><i><font color="Green" title="E63">A42</font></i></a></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" title="c18">H1</font>,<font color="Maroon" title="c21">H2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">H2</font>,<font color="Maroon" title="c18">H1</font>] )
 </span><b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="group_4.html#E2:52_1_5"><i><font color="Green" title="E46">A24</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green" title="E59">A38</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green" title="E64">A43</font></i></a></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" title="c18">H1</font>,<font color="Maroon" title="c21">H2</font>]
 ;<br/><div class="add"><a NAME="E2:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c18">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c21">H2</font>
 <b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><a NAME="E3:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c21">H2</font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_1"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_1"><i><font color="Green" title="E57">A36</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green" title="E59">A38</font></i></a>, <a class="txt" href="group_4.html#E12:52_1_5_2_1_2_1"><i><font color="Green" title="E60">A39</font></i></a>, <a class="txt" href="group_4.html#E15:52_1_5_2_1_2_1"><i><font color="Green" title="E62">A41</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green" title="E64">A43</font></i></a>, <a class="ref" href="group_2.html#T89" title="GROUP_2:th.89">GROUP_2:89</a></i>;<br/><b>hence </b><a NAME="E4:52_1_5_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 <b>by </b><i><a class="txt" href="group_4.html#E13:52_1_5_2_1_2_1"><i><font color="Green" title="E61">A40</font></i></a>, <a class="txt" href="group_4.html#E15:52_1_5_2_1_2_1"><i><font color="Green" title="E62">A41</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green" title="E64">A43</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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" title="c21">H2</font>,<font color="Maroon" title="c18">H1</font>]
 ;<br/><div class="add"><a NAME="E2:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c21">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">H1</font>
 <b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><a NAME="E3:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c18">H1</font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_1"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_1"><i><font color="Green" title="E57">A36</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green" title="E59">A38</font></i></a>, <a class="txt" href="group_4.html#E12:52_1_5_2_1_2_1"><i><font color="Green" title="E60">A39</font></i></a>, <a class="txt" href="group_4.html#E15:52_1_5_2_1_2_1"><i><font color="Green" title="E62">A41</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green" title="E64">A43</font></i></a>, <a class="ref" href="group_2.html#T89" title="GROUP_2:th.89">GROUP_2:89</a></i>;<br/><b>hence </b><a NAME="E4:52_1_5_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2_1"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_1"><i><font color="Green" title="E57">A36</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">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" title="c14">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:52_1_5_2_1_2_2"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_1_5_2_1_2_2"/><i><font color="Green" title="E53">A44</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Z</font>
 <b>and </b><br/><a NAME="E4:52_1_5_2_1_2_2"/><i><font color="Green" title="E54">A45</font></i>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:52_1_5_2_1_2_2"/><i><font color="Green" title="E55">A46</font></i>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">D</font>
 <b>and </b><br/><a NAME="E7:52_1_5_2_1_2_2"/><i><font color="Green" title="E56">A47</font></i>: 
 ex <font color="Olive" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c16">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b1">H</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_2"><i><font color="Green" title="E54">A45</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">H1</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:52_1_5_2_1_2_2"/><i><font color="Green" title="E57">A48</font></i>: 
( <font color="Maroon" title="c17">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c16">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c17">H1</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E7:52_1_5_2_1_2_2"><i><font color="Green" title="E56">A47</font></i></a></i>;<br/><a NAME="E10:52_1_5_2_1_2_2"/>
<font color="Maroon" title="c14">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c17">H1</font>
 <b>by </b><i><a class="txt" href="group_4.html#E3:52_1_5_2_1_2_2"><i><font color="Green" title="E53">A44</font></i></a>, <a class="txt" href="group_4.html#E6:52_1_5_2_1_2_2"><i><font color="Green" title="E55">A46</font></i></a>, <a class="txt" href="group_4.html#E9:52_1_5_2_1_2_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="ref" href="group_2.html#T90" title="GROUP_2:th.90">GROUP_2:90</a></i>;<br/><b>hence </b><a NAME="E11:52_1_5_2_1_2_2"/>
<font color="Maroon" title="c14">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_2"><i><font color="Green" title="E54">A45</font></i></a>, <a class="txt" href="group_4.html#E6:52_1_5_2_1_2_2"><i><font color="Green" title="E55">A46</font></i></a>, <a class="txt" href="group_4.html#E9:52_1_5_2_1_2_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c14">H</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E12:52_1_5_2_1_2"/><i><font color="Green" title="E53">A49</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E</font>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_5_2_1_2"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="group_4.html#E9:52_1_5_2_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="ref" href="group_2.html#T61" title="GROUP_2:th.61">GROUP_2:61</a></i>;<br/><div><i><font color="Green" title="E54">A50</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" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c14">H</font>
 ;<br/><a NAME="E2:52_1_5_2_1_2_3"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E</font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1_5_2_1_2"><i><font color="Green" title="E53">A49</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:52_1_5_2_1_2_3"/><i><font color="Green" title="E55">A51</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Z</font>
 <b>and </b><br/><a NAME="E5:52_1_5_2_1_2_3"/><i><font color="Green" title="E56">A52</font></i>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:52_1_5_2_1_2_3"/><i><font color="Green" title="E57">A53</font></i>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">D</font>
 <b>and </b><br/><a NAME="E8:52_1_5_2_1_2_3"/><i><font color="Green" title="E58">A54</font></i>: 
 ex <font color="Olive" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c16">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b1">H</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2_3"><i><font color="Green" title="E56">A52</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">H1</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E10:52_1_5_2_1_2_3"/><i><font color="Green" title="E59">A55</font></i>: 
( <font color="Maroon" title="c17">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c16">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c17">H1</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_5_2_1_2_3"><i><font color="Green" title="E58">A54</font></i></a></i>;<br/><a NAME="E11:52_1_5_2_1_2_3"/>
not <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c17">H1</font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="group_4.html#E1:52_1_5"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_3"><i><font color="Green" title="E59">A55</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="txt" href="group_4.html#E4:52_1_5_2_1_2_3"><i><font color="Green" title="E55">A51</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_3"><i><font color="Green" title="E57">A53</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_3"><i><font color="Green" title="E59">A55</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E14:52_1_5_2_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c14">H</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">H</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font> <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="group_4.html#E12:52_1_5_2_1_2"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="group_4.html#E13:52_1_5_2_1_2"><i><font color="Green" title="E54">A50</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">y</font>;<br/><b>let </b><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>;<br/><b>assume </b><a NAME="E16:52_1_5_2_1_2"/><i><font color="Green" title="E55">A56</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font>
 ;<br/><a NAME="E17:52_1_5_2_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="group_4.html#E11:52_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">K</font> = <font color="Maroon" title="c17">x</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c19">K</font> = <font color="Maroon" title="c18">K</font>;<br/><b>take </b><font color="Maroon" title="c20">H</font> = <font color="Maroon" title="c14">H</font>;<br/><b>thus </b><a NAME="E19:52_1_5_2_1_2"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">K</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">H</font> )
 ;<br/><a NAME="E20:52_1_5_2_1_2"/>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c19">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">K</font>
 ;<br/><a NAME="E21:52_1_5_2_1_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E16:52_1_5_2_1_2"><i><font color="Green" title="E55">A56</font></i></a></i>;<br/><a NAME="E22:52_1_5_2_1_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E23:52_1_5_2_1_2"/>
<font color="Maroon" title="c19">K</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c20">H</font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1_5_2_1_2"><i><font color="Green" title="E53">A49</font></i></a>, <a class="ref" href="group_2.html#T66" title="GROUP_2:th.66">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" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font><b> such that </b><br/><a NAME="E22:52_1"/><i><font color="Green" title="E45">A57</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font>  st <font color="Maroon" title="c9">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/> not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">s</font>,<font color="Olive" title="b1">y</font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1" title="ORDERS_1:sch.1">ORDERS_1:sch 1</a>(<a class="txt" href="group_4.html#E17:52_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="txt" href="group_4.html#E18:52_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="txt" href="group_4.html#E19:52_1"><i><font color="Green" title="E43">A19</font></i></a>, <a class="txt" href="group_4.html#E20:52_1"><i><font color="Green" title="E44">A22</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c9">s</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_4.html#E16:52_1"><i><font color="Green" title="E40">A16</font></i></a></i>;<br/>
<a NAME="E24:52_1"/>
<font color="Maroon" title="c10">H</font> is <a href="group_4.html#V2" title="GROUP_4:attr.2">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" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c10">H</font>
 
<b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:52_1_6"/>
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c10">H</font> #) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> #)
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_4.html#D7" target="_self" title="GROUP_4:def.7">GROUP_4:def 7</a><br/>

<b>let </b><font color="Maroon" title="c11">K</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:52_1_6"/><i><font color="Green" title="E46">A58</font></i>: 
<font color="Maroon" title="c11">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c10">H</font> #)
 <b>and </b><br/><a NAME="E4:52_1_6"/><i><font color="Green" title="E47">A59</font></i>: 
<font color="Maroon" title="c10">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c11">K</font>
 <b>and </b><br/><a NAME="E5:52_1_6"/><i><font color="Green" title="E48">A60</font></i>: 
<font color="Maroon" title="c11">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> #)
 ;<br/>

<a NAME="E6:52_1_6"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c10">H</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">K</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c10">H</font> )
 
<b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="group_4.html#E4:52_1_6"><i><font color="Green" title="E47">A59</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E7:52_1_6"/><b>then </b><i><font color="Green" title="E49">A61</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">K</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">K</font>
 ;<br/><a NAME="E2:52_1_6_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">K</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E3:52_1_6_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">K</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:52_1_6_1"/><b>then </b><i><font color="Green" title="E50">A62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">K</font>
 <b>by </b><i><a class="txt" href="group_4.html#E7:52_1_6"><i><font color="Green" title="E49">A61</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E5:52_1_6_1"/>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_6_1_1"/><i><font color="Green" title="E51">A63</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_6_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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="E3:52_1_6_1_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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><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" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><a NAME="E2:52_1_6_1_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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="E3:52_1_6_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_6_1_1"><i><font color="Green" title="E51">A63</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:52_1_6_1_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_1_6_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:52_1_6_1"/><b>then </b>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">K</font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_6_1"><i><font color="Green" title="E50">A62</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:52_1_6_1"/><b>then </b>
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c5">B</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T41" target="_self" title="GROUP_4:th.41">Th41</a></i>;<br/><a NAME="E8:52_1_6_1"/><b>then </b>
<font color="Maroon" title="c1">G</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c11">K</font>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="group_4.html#T40" target="_self" title="GROUP_4:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E9:52_1_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_6"><i><font color="Green" title="E48">A60</font></i></a>, <a class="ref" href="group_2.html#T64" title="GROUP_2:th.64">GROUP_2:64</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c11">K</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">X</font> <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_6"><i><font color="Green" title="E49">A61</font></i></a></i>;<br/>
<a NAME="E10:52_1_6"/>
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c12">v</font>]
 
<b>by </b><i><a class="txt" href="group_4.html#E22:52_1"><i><font color="Green" title="E45">A57</font></i></a>, <a class="txt" href="group_4.html#E3:52_1_6"><i><font color="Green" title="E46">A58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:52_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_6"><i><font color="Green" title="E47">A59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:52_1"/><b>then </b>
 <a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c10">H</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T54" target="_self" title="GROUP_4:th.54">Th54</a></i>;<br/>
<a NAME="E26:52_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E27:52_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c10">H</font> &amp; not <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c10">H</font> )
 
<b>by </b><i><a class="txt" href="group_4.html#E1:52_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="group_4.html#E12:52_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E28:52_1"/>
contradiction
 ;<br/>


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

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:52"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : not <font color="Olive" title="b1">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green" title="E33">A64</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E5:52"/><i><font color="Green" title="E34">A65</font></i>: 
not <font color="Maroon" title="c4">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">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" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  holds not <font color="Olive" title="b1">H</font> is <a href="group_4.html#V2" title="GROUP_4:attr.2">maximal</a> or  ex <font color="Olive" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">H</font> is <a href="group_4.html#V2" title="GROUP_4:attr.2">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" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  holds not <font color="Olive" title="b1">H</font> is <a href="group_4.html#V2" title="GROUP_4:attr.2">maximal</a>
 ;<br/><div class="add"><a NAME="E2:52_2_1_1"/><b>then </b>
 <a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_4.html#D8" target="_self" title="GROUP_4:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:52_2_1_1"/>
<font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52"><i><font color="Green" title="E33">A64</font></i></a></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" title="E35">A66</font></i>: 
 ex <font color="Olive" title="b1">H</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">H</font> is <a href="group_4.html#V2" title="GROUP_4:attr.2">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" title="c5">H</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:52_2_1_2_1"/><i><font color="Green" title="E36">A67</font></i>: 
<font color="Maroon" title="c5">H</font> is <a href="group_4.html#V2" title="GROUP_4:attr.2">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" title="E37">A68</font></i>: 
not <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">H</font>
 ;<br/><a NAME="E2:52_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E38">A69</font></i>: 
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2_1"><i><font color="Green" title="E36">A67</font></i></a>, <a class="ref" href="group_4.html#T48" target="_self" title="GROUP_4:th.48">Th48</a></i>;<br/><a NAME="E3:52_2_1_2_1_1"/><i><font color="Green" title="E39">A70</font></i>: 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:52_2_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:52_2_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2_1_1"><i><font color="Green" title="E37">A68</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">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" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font>
<b>by </b><i><a class="txt" href="group_4.html#E3:52_2_1_2_1_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="ref" href="xboole_1.html#T83" title="XBOOLE_1:th.83">XBOOLE_1:83</a></i>
;<br/><a NAME="E5:52_2_1_2_1_1"/><b>then </b>
(  <a href="group_4.html#K5" title="GROUP_4:func.5">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" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c5">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2_1"><i><font color="Green" title="E36">A67</font></i></a>, <a class="ref" href="group_4.html#D7" target="_self" title="GROUP_4:def.7">Def7</a>, <a class="ref" href="group_4.html#T40" target="_self" title="GROUP_4:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E6:52_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E5:52"><i><font color="Green" title="E34">A65</font></i></a>, <a class="txt" href="group_4.html#E2:52_2_1_2_1_1"><i><font color="Green" title="E38">A69</font></i></a>, <a class="ref" href="group_4.html#D6" target="_self" title="GROUP_4:def.6">Def6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_2_1_2_1"/>
<font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">H</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:52_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2"><i><font color="Green" title="E35">A66</font></i></a>, <a class="ref" href="group_4.html#T52" target="_self" title="GROUP_4:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E4:52_2_1_2"/>
<font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52"><i><font color="Green" title="E33">A64</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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" title="c3">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 <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ;<br/>


</div>
