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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">B</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">M</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">J</font> = <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> ;<br/>
<a NAME="E2:35"/><i><font color="Green" title="E24">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/>
<a NAME="E5:35_1"/>
<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a>  <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="group_6.html#D1" target="_self" title="GROUP_6:def.1">Def1</a></i>;<br/>
<a NAME="E6:35_1"/><b>then </b>
( <font color="Maroon" title="c7">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c8">b</font> )
 
<b>by </b><i><a class="ref" href="group_6.html#T2" target="_self" title="GROUP_6:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:35_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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E3:35_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="group_6.html#T16" target="_self" title="GROUP_6:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c6">g</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">f</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">X</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E3:35"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="group_6.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E4:35"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:35_2"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c11">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_2"/><i><font color="Green" title="E27">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:35_2"/><i><font color="Green" title="E28">A7</font></i>: 
( <font color="Maroon" title="c10">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">z</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="group_6.html#E1:35_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="group_6.html#E3:35_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E6:35_2"/><i><font color="Green" title="E29">A8</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c12">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E8:35_2"/><i><font color="Green" title="E30">A9</font></i>: 
( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c13">b</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E4:35_2"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">W1</font> = <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c15">W2</font> = <font color="Maroon" title="c11">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <b>by </b><i><a class="txt" href="group_6.html#E6:35_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="group_6.html#E8:35_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="group_6.html#T16" target="_self" title="GROUP_6:th.16">Th16</a></i>;<br/><i><font color="Green" title="E31">A10</font></i>: the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">W1</font>,<font color="Maroon" title="c15">W2</font>
<b>by </b><i><a class="txt" href="group_6.html#E3:35_2"><i><font color="Green" title="E27">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c13">b</font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E6:35_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="group_6.html#E8:35_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c13">b</font>
<b>by </b><i><a class="ref" href="group_3.html#T12" title="GROUP_3:th.12">GROUP_3:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c13">b</font>
<b>by </b><i><a class="ref" href="group_4.html#T60" title="GROUP_4:th.60">GROUP_4:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c13">b</font>
<b>by </b><i><a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></i>
<br/>.= 
<font color="Maroon" title="c12">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c13">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T128" title="GROUP_2:th.128">GROUP_2:128</a></i>
<br/>.= 
<font color="Maroon" title="c12">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c16">a'</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c17">b'</font> = <font color="Maroon" title="c13">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/><a NAME="E12:35_2"/><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a>  <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="group_6.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="group_6.html#D1" target="_self" title="GROUP_6:def.1">Def1</a></i>;<br/><a NAME="E13:35_2"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c16">a'</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c17">b'</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E6:35_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="group_6.html#E8:35_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="group_6.html#T2" target="_self" title="GROUP_6:th.2">Th2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">V1</font> = <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c19">V2</font> = <font color="Maroon" title="c11">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c3">M</font> <b>by </b><i><a class="ref" href="group_6.html#T16" target="_self" title="GROUP_6:th.16">Th16</a></i>;<br/><i><font color="Green" title="E34">A13</font></i>: the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">V1</font>,<font color="Maroon" title="c19">V2</font>
<b>by </b><i><a class="txt" href="group_6.html#E3:35_2"><i><font color="Green" title="E27">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c17">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E13:35_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">a'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c17">b'</font>
<b>by </b><i><a class="ref" href="group_3.html#T12" title="GROUP_3:th.12">GROUP_3:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a'</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c17">b'</font>
<b>by </b><i><a class="ref" href="group_4.html#T60" title="GROUP_4:th.60">GROUP_4:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c17">b'</font>
<b>by </b><i><a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></i>
<br/>.= 
<font color="Maroon" title="c16">a'</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c17">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T128" title="GROUP_2:th.128">GROUP_2:128</a></i>
<br/>.= 
<font color="Maroon" title="c16">a'</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">b'</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">M</font>
<b>by </b><i><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></i>
;<br/><a NAME="E16:35_2"/>
<font color="Maroon" title="c16">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E17:35_2"/>
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="group_6.html#E10:35_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="group_6.html#E12:35_2"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="group_6.html#E15:35_2"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="group_6.html#T2" target="_self" title="GROUP_6:th.2">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:35"/><b>then </b>
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_6.html#E4:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E7:35"/>
<font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="group_6.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>


</div>
