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

<b>set </b><font color="Maroon" title="c5">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 ;<br/><a NAME="E2:20_1_1_1"/><b>then </b>
 ex <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 <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )
 ;<br/><b>hence </b><a NAME="E3:20_1_1_1"/>
<font color="Maroon" title="c6">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>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> </span>}</span>  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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:20_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></i>;<br/>
<a NAME="E4:20_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></i>;<br/>
<a NAME="E5:20_1_1"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
;<br/>
<a NAME="E6:20_1_1"/><i><font color="Green" title="E14">A3</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> <font color="Maroon" title="c6">X</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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="c7">g1</font>, <font color="Maroon" title="c8">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_2"/>
<font color="Maroon" title="c7">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>

<a NAME="E2:20_1_1_2"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
 ex <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 <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )
 
;<br/>
<b>assume </b><a NAME="E3:20_1_1_2"/>
<font color="Maroon" title="c8">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>

<a NAME="E4:20_1_1_2"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
 ex <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 <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )
 
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g2</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c8">g2</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="txt" href="group_10.html#E2:20_1_1_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="group_10.html#E4:20_1_1_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>
;<br/>
<b>hence </b><a NAME="E6:20_1_1_2"/>
<font color="Maroon" title="c7">g1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:20_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> <font color="Maroon" title="c6">X</font> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<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="c7">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_3"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>

<a NAME="E2:20_1_1_3"/><b>then </b>
 ex <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 <br/>( <font color="Olive" title="b1">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g'</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )
 
;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c7">g</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">g</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="txt" href="group_10.html#E3:20_1_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E4:20_1_1_3"/>
<font color="Maroon" title="c7">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:20_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">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> <span class="p1">{<span class="default"> <font color="Olive" title="b2">g</font> where <font color="Olive" title="b2">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="group_10.html#E5:20_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="group_10.html#E6:20_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="group_2.html#T61" title="GROUP_2:th.61">GROUP_2:61</a></i>;<br/>


</div>
