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

<b>set </b><font color="Maroon" title="c2">cG</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> ;<br/>
<a NAME="E1:20_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> )</span><br/>

<b>assume </b><a NAME="E1:20_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><br/>

<a NAME="E2:20_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font>
 
;<br/>
<b>hence </b><a NAME="E3:20_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ) &amp; ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  )</span>
<div class="add"><b>assume </b><a NAME="E1:20_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><br/><b>then consider </b><font color="Maroon" title="c5">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_1_1_2_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <b>and </b><br/><a NAME="E4:20_1_1_2_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E5:20_1_1_2_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:20_1_1_2_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E6:20_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:20_1_1_2_1"><i><font color="Green" title="E17">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:20_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> </span><br/><b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ;<br/><b>reconsider </b><font color="Maroon" title="c6">S</font> =  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c5">x'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E5:20_1_1_2"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> 
 ;<br/><a NAME="E6:20_1_1_2"/>
<font color="Maroon" title="c5">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c5">x'</font>
 <b>by </b><i><a class="ref" href="group_3.html#T98" title="GROUP_3:th.98">GROUP_3:98</a></i>;<br/><b>hence </b><a NAME="E7:20_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:20_1_1_2"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:20_1_1"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  <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="c1">G</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  implies ( <font color="Maroon" title="c4">A</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>  &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : verum </span>}</span>  &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B</font> holds <br/>not <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">B</font> ) ) )</span><br/><b>assume </b><a NAME="E1:20_1_1_3"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</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>  &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : verum </span>}</span>  &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B</font> holds <br/>not <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">B</font> ) )</span><br/><a NAME="E2:20_1_1_3"/><b>then </b><i><font color="Green" title="E18">A9</font></i>: 
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font>
 ;<br/><b>consider </b><font color="Maroon" title="c5">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:20_1_1_3"/><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:20_1_1_3"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E5:20_1_1_3"/>
<font color="Maroon" title="c4">A</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="weddwitt.html#E2:20_1_1_3"><i><font color="Green" title="E18">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : verum </span>}</span>  &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B</font> holds <br/>not <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">B</font></span><br/><b>let </b><font color="Maroon" title="c6">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">B</font> implies not <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">B</font> )</span><br/><b>assume </b><a NAME="E6:20_1_1_3"/>
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">B</font> implies not <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">B</font> )</span><br/><a NAME="E7:20_1_1_3"/><b>then </b><i><font color="Green" title="E20">A12</font></i>: 
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font>
 ;<br/><b>consider </b><font color="Maroon" title="c7">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:20_1_1_3"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E7:20_1_1_3"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/><b>assume </b><a NAME="E10:20_1_1_3"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">B</font></span><br/><b>assume </b><a NAME="E11:20_1_1_3"/>
<font color="Maroon" title="c4">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:20_1_1_3"/><i><font color="Green" title="E23">A15</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="weddwitt.html#E13:20_1_1_3"><i><font color="Green" title="E23">A15</font></i></a></i>;<br/><a NAME="E15:20_1_1_3"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">a</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:20_1_1_3"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="weddwitt.html#E13:20_1_1_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="group_3.html#T96" title="GROUP_3:th.96">GROUP_3:96</a></i>;<br/><a NAME="E16:20_1_1_3"/>
<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E9:20_1_1_3"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="weddwitt.html#E13:20_1_1_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="group_3.html#T96" title="GROUP_3:th.96">GROUP_3:96</a></i>;<br/><a NAME="E17:20_1_1_3"/><b>then </b><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E15:20_1_1_3"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="group_3.html#T91" title="GROUP_3:th.91">GROUP_3:91</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> implies <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> implies <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:20_1_1_3_1_1"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font></span><br/><b>then reconsider </b><font color="Maroon" title="c11">c'</font> = <font color="Maroon" title="c10">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:20_1_1_3_1_1"/>
<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c5">a</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:20_1_1_3"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="weddwitt.html#E1:20_1_1_3_1_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="group_3.html#T96" title="GROUP_3:th.96">GROUP_3:96</a></i>;<br/><a NAME="E4:20_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E17:20_1_1_3"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="group_3.html#T91" title="GROUP_3:th.91">GROUP_3:91</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1_3_1_1"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E9:20_1_1_3"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="group_3.html#T96" title="GROUP_3:th.96">GROUP_3:96</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:20_1_1_3_1"/><i><font color="Green" title="E26">A19</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/><b>then reconsider </b><font color="Maroon" title="c11">c'</font> = <font color="Maroon" title="c10">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E4:20_1_1_3_1"/>
<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E9:20_1_1_3"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="weddwitt.html#E2:20_1_1_3_1"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="group_3.html#T96" title="GROUP_3:th.96">GROUP_3:96</a></i>;<br/><a NAME="E5:20_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c5">a</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E17:20_1_1_3"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="group_3.html#T91" title="GROUP_3:th.91">GROUP_3:91</a></i>;<br/><b>hence </b><a NAME="E6:20_1_1_3_1"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:20_1_1_3"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="group_3.html#T96" title="GROUP_3:th.96">GROUP_3:96</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E19:20_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E10:20_1_1_3"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:20_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span> 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> : verum </span>}</span>  is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:20_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="weddwitt.html#E3:20_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
