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

<a NAME="E1:20_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:20_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:20_1_1_1"/>
<span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_1_1"/>
<span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> holds <br/> <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:20_1_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X1</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E6:20_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:20_1_1_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:20_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Olive">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">g2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:20_1_1_3"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> )
 ;<br/>

<a NAME="E2:20_1_1_3"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_3_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:20_1_1_3_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="latsubgr.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:20_1_1_3_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> )
 <b>by </b><i><a class="ref" href="latsubgr.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X1</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">g1</font>, <font color="Maroon">h2</font> = <font color="Maroon">g2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E9:20_1_1_3_1"/>
( <font color="Maroon">h1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">h2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:20_1_1_3_1"/><b>then </b>
<font color="Maroon">h1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_3_1"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_3"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">CG</font> = <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E6:20_1_1"/>
for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">CG</font> holds <br/><font color="Olive">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">CG</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:20_1_1_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">CG</font>
 ;<br/>

<a NAME="E2:20_1_1_4"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_4_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:20_1_1_4_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="latsubgr.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:20_1_1_4_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> )
 <b>by </b><i><a class="ref" href="latsubgr.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X1</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_4_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E9:20_1_1_4_1"/>
<font color="Maroon">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:20_1_1_4_1"/><b>then </b>
<font color="Maroon">h</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_4_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_4"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">CG</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>


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