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

<a NAME="E1:20_1_1"/><i><font color="Green" title="E10">A1</font></i>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</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> 
 
<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" title="c3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:20_1_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:20_1_1_1"/>
<span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E2:20_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">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" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</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><b>end;</b></div>
<a NAME="E2:20_1_1"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1_2"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c3">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<b>consider </b><font color="Maroon" title="c5">X1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:20_1_1_2"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">X1</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:20_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">X1</font> = <font color="Maroon" title="c5">X1</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> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E6:20_1_1_2"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c4">X</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="c6">X1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E4:20_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="latsubgr.html#D1" target="_self" title="LATSUBGR:def.1">Def1</a></i>;<br/>
<a NAME="E7:20_1_1_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">X1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55" title="GROUP_2:th.55">GROUP_2:55</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E6:20_1_1_2"><i><font color="Green" title="E13">A5</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="E3:20_1_1"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font></span>)</span> <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="latsubgr.html#E1:20_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:20_1_1"/><i><font color="Green" title="E12">A7</font></i>: 
for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</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="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font></span>)</span> holds <br/><font color="Olive" title="b1">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">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" title="c3">g1</font>, <font color="Maroon" title="c4">g2</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="E1:20_1_1_3"/><i><font color="Green" title="E13">A8</font></i>: 
( <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font></span>)</span> )
 ;<br/>

<a NAME="E2:20_1_1_3"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c3">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1_3_1"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E2:20_1_1_3_1"/><b>then </b><i><font color="Green" title="E15">A10</font></i>: 
( <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> )
 
<b>by </b><i><a class="txt" href="latsubgr.html#E1:20_1_1_3"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> 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="txt" href="latsubgr.html#E1:20_1_1_3_1"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">X1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:20_1_1_3_1"/><i><font color="Green" title="E16">A11</font></i>: 
( <font color="Maroon" title="c7">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">X1</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:20_1_1_3_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">X1</font> = <font color="Maroon" title="c7">X1</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> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_3_1"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c6">X</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="c8">X1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E5:20_1_1_3_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="latsubgr.html#D1" target="_self" title="LATSUBGR:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h1</font> = <font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c10">h2</font> = <font color="Maroon" title="c4">g2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E9:20_1_1_3_1"/>
( <font color="Maroon" title="c9">h1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">X1</font> &amp; <font color="Maroon" title="c10">h2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">X1</font> )
 
<b>by </b><i><a class="txt" href="latsubgr.html#E2:20_1_1_3_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="latsubgr.html#E7:20_1_1_3_1"><i><font color="Green" title="E17">A12</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="E10:20_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c9">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">h2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">X1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_3_1"/>
<font color="Maroon" title="c3">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E7:20_1_1_3_1"><i><font color="Green" title="E17">A12</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>hence </b><a NAME="E3:20_1_1_3"/>
<font color="Maroon" title="c3">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:20_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c3">CG</font> = <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E6:20_1_1"/>
for <font color="Olive" title="b1">g</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="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">CG</font> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">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" title="c4">g</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="E1:20_1_1_4"/><i><font color="Green" title="E13">A13</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">CG</font>
 ;<br/>

<a NAME="E2:20_1_1_4"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1_4_1"/><i><font color="Green" title="E14">A14</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E2:20_1_1_4_1"/><b>then </b><i><font color="Green" title="E15">A15</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E1:20_1_1_4"><i><font color="Green" title="E13">A13</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> 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="txt" href="latsubgr.html#E1:20_1_1_4_1"><i><font color="Green" title="E14">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">X1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:20_1_1_4_1"/><i><font color="Green" title="E16">A16</font></i>: 
( <font color="Maroon" title="c7">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">X1</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:20_1_1_4_1"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">X1</font> = <font color="Maroon" title="c7">X1</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> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_4_1"/><i><font color="Green" title="E17">A17</font></i>: 
<font color="Maroon" title="c6">X</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="c8">X1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E5:20_1_1_4_1"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="latsubgr.html#D1" target="_self" title="LATSUBGR:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c4">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E9:20_1_1_4_1"/>
<font color="Maroon" title="c9">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">X1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E2:20_1_1_4_1"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="latsubgr.html#E7:20_1_1_4_1"><i><font color="Green" title="E17">A17</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="E10:20_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c9">h</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">X1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_4_1"/>
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E7:20_1_1_4_1"><i><font color="Green" title="E17">A17</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>hence </b><a NAME="E3:20_1_1_4"/>
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">CG</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:20_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">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="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> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" title="LATSUBGR:func.1">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:20_1_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="latsubgr.html#E4:20_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="group_2.html#T61" title="GROUP_2:th.61">GROUP_2:61</a></i>;<br/>


</div>
