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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> (  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">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="b2">b</font> iff  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b2">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">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="c3">b</font> iff  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:100"/>
(  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">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="c3">b</font> implies  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font> implies  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">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="c3">b</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:100_1"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">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="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E2:100_1"/>
<font color="Maroon" title="c2">a</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="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T96" target="_self" title="GROUP_3:th.96">Th96</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:100_1"/>
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:100"/>
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">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="c3">b</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:100"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c4">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="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E5:100"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<font color="Maroon" title="c4">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="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E59">A2</font></span></a></span>;<br/>
<a NAME="E7:100"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="group_3.html#T96" target="_self" title="GROUP_3:th.96">Th96</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:100"/>
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">y</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="c2">a</font> or <font color="Maroon" title="c6">y</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="c3">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:100_2"/>
<font color="Maroon" title="c6">y</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="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">y</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="c3">b</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:100_2"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">and </span><br/><a NAME="E4:100_2"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">g</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T95" target="_self" title="GROUP_3:th.95">Th95</a></span>;<br/>
<a NAME="E5:100_2"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="group_3.html#T96" target="_self" title="GROUP_3:th.96">Th96</a></span>;<br/>
<a NAME="E6:100_2"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="group_3.html#T96" target="_self" title="GROUP_3:th.96">Th96</a></span>;<br/>
<a NAME="E7:100_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">g</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100_2"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="group_3.html#T91" target="_self" title="GROUP_3:th.91">Th91</a></span>;<br/>
<a NAME="E8:100_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">g</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:100_2"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="ref" href="group_3.html#T91" target="_self" title="GROUP_3:th.91">Th91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:100_2"/>
<font color="Maroon" title="c6">y</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="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100_2"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="group_3.html#T95" target="_self" title="GROUP_3:th.95">Th95</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">y</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="c3">b</font> or <font color="Maroon" title="c6">y</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="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:100"/>
<font color="Maroon" title="c6">y</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="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">y</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="c2">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E11:100"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">and </span><br/><a NAME="E12:100"/><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">g</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T95" target="_self" title="GROUP_3:th.95">Th95</a></span>;<br/>
<a NAME="E13:100"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="group_3.html#T96" target="_self" title="GROUP_3:th.96">Th96</a></span>;<br/>
<a NAME="E14:100"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">g</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:100"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="group_3.html#T91" target="_self" title="GROUP_3:th.91">Th91</a></span>;<br/>
<a NAME="E15:100"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">g</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:100"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="group_3.html#T91" target="_self" title="GROUP_3:th.91">Th91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:100"/>
<font color="Maroon" title="c6">y</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="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:100"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="group_3.html#T95" target="_self" title="GROUP_3:th.95">Th95</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
