<?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">B</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><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="E1:34"/><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">B</font>
 ;<br/>

<b>set </b><font color="Maroon">I</font> = <font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">J</font> = <font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> ;<br/>
<a NAME="E2:34"/><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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="E1:34_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</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/>
<a NAME="E5:34_1"/>
<font color="Maroon">B</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>, </i>;<br/>
<a NAME="E6:34_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:34_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">M</font> &amp; the <a href="struct_0.html#U1">carrier</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> <a href="hidden.html#R1">=</a>  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">g</font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">f</font> = the <a href="group_1.html#U1">mult</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>set </b><font color="Maroon">X</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E3:34"/>
(  <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </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="E1:34_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_2"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:34_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:34_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E8:34_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">y</font>, <font color="Maroon">W2</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <b>by </b><i>, , </i>;<br/><i><font color="Green">E112</font></i>: the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b</font>
<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">a</font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_4.html#K7">*</a> <span class="p3">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_4.html#T60">GROUP_4:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_4.html#T58">GROUP_4:58</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T128">GROUP_2:128</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</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/><a NAME="E12:34_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">B</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>, </i>;<br/><a NAME="E13:34_2"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">y</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">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">y</font>, <font color="Maroon">V2</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">M</font> <b>by </b><i/>;<br/><i><font color="Green">E117</font></i>: the <a href="group_1.html#U1">mult</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="group_1.html#U1">mult</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">V1</font>,<font color="Maroon">V2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</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">b'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="group_2.html#K10">*</a> <font color="Maroon">M</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b'</font>
<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">a'</font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_4.html#K7">*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="group_4.html#T60">GROUP_4:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="group_4.html#T58">GROUP_4:58</a></i>
<br/>.= 
<font color="Maroon">a'</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T128">GROUP_2:128</a></i>
<br/>.= 
<font color="Maroon">a'</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
;<br/><a NAME="E16:34_2"/>
<font color="Maroon">a'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E17:34_2"/>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E6:34"/><b>then </b>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</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> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E7:34"/>
<font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><font color="Maroon">B</font>,<font color="Maroon">M</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> is    <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_2.html#D5">GROUP_2:def 5</a></i>;<br/>


</div>
