<?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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">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/> (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b2">H2</font> iff  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b2">H2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> iff  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )</span><br/>



<span class="kw">thus </span><a NAME="E1:116"/>
(  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> implies  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> implies  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:116_1"/><span class="lab"><font color="Green" title="E78">A1</font></span>: 
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font></span><br/>

<a NAME="E2:116_1"/>
<font color="Maroon" title="c2">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T109" target="_self" title="GROUP_3:th.109">Th130</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:116_1"/>
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116_1"><span class="lab"><font color="Green" title="E78">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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:116"/>
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</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="E4:116"/><span class="lab"><font color="Green" title="E78">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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font>
 <span class="kw">and </span><br/><a NAME="E5:116"/><span class="lab"><font color="Green" title="E79">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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</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="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="group_3.html#T106" target="_self" title="GROUP_3:th.106">Th127</a></span>;<br/>
<a NAME="E7:116"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th128</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:116"/>
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
  <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_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="firebrick">:: 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="firebrick">::  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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">H3</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:116_2"/><span class="lab"><font color="Green" title="E81">A5</font></span>: 
<font color="Maroon" title="c7">H3</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="E4:116_2"/><span class="lab"><font color="Green" title="E82">A6</font></span>: 
<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></span>;<br/>
<a NAME="E5:116_2"/><span class="lab"><font color="Green" title="E83">A7</font></span>: 
<font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th128</a></span>;<br/>
<a NAME="E6:116_2"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">H1</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th128</a></span>;<br/>
<a NAME="E7:116_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_2"><span class="lab"><font color="Green" title="E82">A6</font></span></a>, <a class="ref" href="group_3.html#T105" target="_self" title="GROUP_3:th.105">Th124</a></span>;<br/>
<a NAME="E8:116_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116_2"><span class="lab"><font color="Green" title="E83">A7</font></span></a>, <a class="ref" href="group_3.html#T105" target="_self" title="GROUP_3:th.105">Th124</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:116_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_2"><span class="lab"><font color="Green" title="E81">A5</font></span></a>, <a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="firebrick">:: 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="firebrick">::  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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:116"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">H3</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E11:116"/><span class="lab"><font color="Green" title="E81">A8</font></span>: 
<font color="Maroon" title="c7">H3</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="E12:116"/><span class="lab"><font color="Green" title="E82">A9</font></span>: 
<font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></span>;<br/>
<a NAME="E13:116"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H2</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th128</a></span>;<br/>
<a NAME="E14:116"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:116"><span class="lab"><font color="Green" title="E82">A9</font></span></a>, <a class="ref" href="group_3.html#T105" target="_self" title="GROUP_3:th.105">Th124</a></span>;<br/>
<a NAME="E15:116"/><span class="kw">then </span>
<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:116"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="group_3.html#T105" target="_self" title="GROUP_3:th.105">Th124</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:116"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:116"><span class="lab"><font color="Green" title="E81">A8</font></span></a>, <a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
