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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">N</font> be  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>, <font color="Maroon">M</font> be  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">M</font> is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">N</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a>  <a href="group_2.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">N</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><font color="Maroon">N</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> as    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">M</font> as  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span>;<br/><b>thus </b><a NAME="E1:35_1"/>
<font color="Maroon">S</font> <a href="group_2.html#K12">*</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font> <a href="group_2.html#K13">*</a> <font color="Maroon">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><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="E1:35_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="group_2.html#K12">*</a> <font color="Maroon">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon">T</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span><b> such that </b><br/><a NAME="E3:35_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="group_1.html#K1">*</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E4:35_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">T</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="group_2.html#T125">GROUP_2:125</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:35_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E8:35_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">I</font> &amp; <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<b>set </b><font color="Maroon">e</font> = <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>;<br/>
<a NAME="E10:35_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E11:35_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="group_3.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:35_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="group_5.html#T3">GROUP_5:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">I</font> <a href="group_2.html#K13">*</a> <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">J</font> <b>by </b><i>, </i>;<br/>
<a NAME="E15:35_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<a NAME="E17:35_1_1"/><i><font color="Green">E110</font></i>: 
(  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp;  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> &amp; <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font> &amp; <font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="group_2.html#K13">*</a> <font color="Maroon">f</font> &amp;  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> )
 
<b>by </b><i/>;<br/>
<b>then </b><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T43">GROUP_2:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T129">GROUP_2:129</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">M</font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T12">GROUP_3:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T118">GROUP_2:118</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T129">GROUP_2:129</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T129">GROUP_2:129</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T128">GROUP_2:128</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T13">GROUP_3:13</a></i>
<br/>.= 
<font color="Maroon">V</font> <a href="group_1.html#K1">*</a> <font color="Maroon">S</font>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E19:35_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="group_2.html#K13">*</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">N</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><font color="Maroon">N</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> is  <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="group_3.html#T141">GROUP_3:141</a></i>;<br/>


</div>
