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

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">B</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon">G</font> <a href="group_5.html#K9">`</a>  is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">B</font> implies for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="group_5.html#K2">.]</a></span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">G</font> <a href="group_5.html#K9">`</a>  is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">B</font>
 ;<br/>

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

<a NAME="E2:9_1"/>
<span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="group_5.html#K2">.]</a></span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G</font> <a href="group_5.html#K9">`</a> 
 
<b>by </b><i><a class="ref" href="group_5.html#T84">GROUP_5:84</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1"/>
<span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="group_5.html#K2">.]</a></span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T49">GROUP_2:49</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:9"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="group_5.html#K2">.]</a></span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> = <span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="group_5.html#K2">.]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>from </b><i><a class="ref" href="domain_1.html#S9">DOMAIN_1:sch 9</a>();<br/></i>
<a NAME="E4:9"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:9_2"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="group_5.html#K2">.]</a></span>
 
;<br/>
<a NAME="E3:9_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</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="E5:9"/><b>then </b>
 <a href="group_4.html#K5">gr</a> <font color="Maroon">X</font> is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="group_4.html#D5">GROUP_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:9"/>
<font color="Maroon">G</font> <a href="group_5.html#K9">`</a>  is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


</div>
