<?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>, <font color="Maroon">H</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">h</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/>



<b>assume </b><a NAME="E1:19"/>
<font color="Maroon">G</font> is  <a href="grsolv_1.html#V1" target="_self">solvable</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:19"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/>for <font color="Olive">G1</font>, <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">G2</font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">G1</font> &amp; ( for <font color="Olive">N</font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">G1</font>  st <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Olive">G2</font> holds <br/><font color="Olive">G1</font> <a href="group_6.html#K6">./.</a> <font color="Olive">N</font> is <a href="group_1.html#V7">commutative</a> ) ) ) )
 <b>by </b><i/>;<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> ex <font color="Olive">J</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Olive">J</font> );<br/>
<a NAME="E4:19"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E4:19_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:19_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>
<a NAME="E7:19_1"/>
<font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font> is  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of  <a href="group_6.html#K12">Image</a> <font color="Maroon">h</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:19_1"/><b>then </b>
<font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span><b> such that </b><br/><a NAME="E10:19_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font>
 ;<br/>
<b>thus </b><a NAME="E11:19_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">x</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">z</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span><b> such that </b><br/><a NAME="E6:19"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<a NAME="E7:19"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E54</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E10:19"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E11:19"/><b>then </b><i><font color="Green">E55</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:19"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">A</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:19_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E3:19_3"/>
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>

<a NAME="E4:19_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:19_3"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> = 
 <a href="group_2.html#K5">(1).</a> <font color="Maroon">H</font>
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="group_2.html#K5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T75">GROUP_2:75</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E13:19"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> &amp; <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> holds <br/>for <font color="Olive">G1</font>, <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of  <a href="group_6.html#K12">Image</a> <font color="Maroon">h</font>  st <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">G2</font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">G1</font> &amp; ( for <font color="Olive">N</font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">G1</font>  st <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Olive">G2</font> holds <br/><font color="Olive">G1</font> <a href="group_6.html#K6">./.</a> <font color="Olive">N</font> is <a href="group_1.html#V7">commutative</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19_4"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> &amp; <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> )
 ;<br/>

<b>let </b><font color="Maroon">G1</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of  <a href="group_6.html#K12">Image</a> <font color="Maroon">h</font>, <font color="Maroon">G2</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of  <a href="group_6.html#K12">Image</a> <font color="Maroon">h</font>;<br/>

<b>assume </b><a NAME="E2:19_4"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:19_4"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">G1</font> <a href="group_2.html#R1">=</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">B</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:19_4"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">G2</font> <a href="group_2.html#R1">=</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E7:19_4"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">B</font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E8:19_4"/>
for <font color="Olive">N</font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">G1</font>  st <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> holds <br/><font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Olive">N</font> is <a href="group_1.html#V7">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N</font> be  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">G1</font>;<br/>

<b>assume </b><a NAME="E1:19_4_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font></span>)</span>;<br/><a NAME="E1:19_4_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G1</font> <a href="group_6.html#K6">./.</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="E2:19_4_1_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">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font></span>)</span> &amp; <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">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font></span>)</span> )
 ;<br/><a NAME="E3:19_4_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4">Cosets</a> <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4">Cosets</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T22">GROUP_6:22</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E5:19_4_1_1"/><i><font color="Green">E71</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> <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="group_6.html#T28">GROUP_6:28</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E7:19_4_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="group_6.html#T28">GROUP_6:28</a></i>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="grsolv_1.html#D1" target="_self">Def1</a>, <a class="ref" href="group_6.html#T17">GROUP_6:17</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="grsolv_1.html#D1" target="_self">Def1</a>, <a class="ref" href="group_6.html#T17">GROUP_6:17</a></i>;<br/><a NAME="E10:19_4_1_1"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E11:19_4_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> )
 ;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E13:19_4_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/><b>consider </b><font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E15:19_4_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</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>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b1</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="E18:19_4_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E19:19_4_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><i><font color="Green">E111</font></i>: <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E112</font></i>: <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i/>
;<br/><b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E37</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E4:19"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E24:19_4_1_1"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E25:19_4_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">y2</font>
 ;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:19_4_1_1_3_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> )
 ;<br/><b>reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">z1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">z22</font> = <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:19_4_1_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E41</font></i></a>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">x2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">y2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_3_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_4_1_1_3_2"/><i><font color="Green">E121</font></i>: 
( <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">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="grsolv_1.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:19_4_1_1_3_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="grsolv_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:19_4_1_1_3_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>then reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> ;<br/><b>reconsider </b><font color="Maroon">z22</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/><font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font></span>)</span>
<b>by </b><i><a class="ref" href="grsolv_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
;<br/><b>hence </b><a NAME="E11:19_4_1_1_3_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">y2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">y2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_3"/>
<font color="Maroon">x2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:19_4_1_1"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E27:19_4_1_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">y2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">x2</font>
 ;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:19_4_1_1_4_1"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> )
 ;<br/><b>reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">z1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">z22</font> = <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i/>;<br/><a NAME="E6:19_4_1_1_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_4"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">y2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">x2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_4_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_4_1_1_4_2"/><i><font color="Green">E128</font></i>: 
( <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">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:19_4_1_1_4_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/><a NAME="E7:19_4_1_1_4_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>then reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> ;<br/><b>reconsider </b><font color="Maroon">z22</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/><font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
;<br/><b>hence </b><a NAME="E11:19_4_1_1_4_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">y2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_4"/>
<font color="Maroon">y2</font> <a href="group_2.html#K2">*</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:19_4_1_1"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">B</font> as  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">A</font> <b>by </b><i>, , , , </i>;<br/>
<b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a1</font>, <font color="Maroon">b2</font> = <font color="Maroon">b1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E3:19_4_1_1_5"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E4:19_4_1_1_5"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> holds  the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</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">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span>;<br/>

<a NAME="E1:19_4_1_1_5_1"/>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i>, <a class="ref" href="group_3.html#T2">GROUP_3:2</a></i>;<br/>
<b>hence </b><a NAME="E2:19_4_1_1_5_1"/>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</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">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_4_1_1_5"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">K</font> &amp; <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">K</font> )
 
<b>by </b><i><a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>;<br/>
<a NAME="E6:19_4_1_1_5"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">a2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">K</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T27">GROUP_6:27</a></i>;<br/>
<a NAME="E7:19_4_1_1_5"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">b2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">K</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T27">GROUP_6:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a12</font> = <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font>, <font color="Maroon">b12</font> = <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4">Cosets</a> <font color="Maroon">K</font> <b>by </b><i>, <a class="ref" href="group_6.html#T16">GROUP_6:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a11</font> = <font color="Maroon">a12</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <b>by </b><i>, <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b11</font> = <font color="Maroon">b12</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <b>by </b><i>, <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<i><font color="Green">E144</font></i>: <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a11</font>,<font color="Maroon">b11</font>
<b>by </b><i><a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a12</font>,<font color="Maroon">b12</font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b11</font>,<font color="Maroon">a11</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b12</font>,<font color="Maroon">a12</font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon">B1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A1</font>
<b>by </b><i><a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
;<br/>
<a NAME="E14:19_4_1_1_5"/><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_3_1"><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="E1:19_4_1_1_5_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_3_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E4:19_4_1_1_5_3_1"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green">E51</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">z1</font>, <font color="Maroon">z22</font> = <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E6:19_4_1_1_5_3_1"/>
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_5_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_5_3"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_3_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="E1:19_4_1_1_5_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font>
 ;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_3_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">z1</font>, <font color="Maroon">z22</font> = <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E5:19_4_1_1_5_3_2"/>
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E6:19_4_1_1_5_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_5_3"/><b>then </b>
<font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_5_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:19_4_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_4_1"><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="E1:19_4_1_1_5_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_4_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E4:19_4_1_1_5_4_1"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green">E51</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">z1</font>, <font color="Maroon">z22</font> = <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E6:19_4_1_1_5_4_1"/>
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_5_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_5_4"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_4_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="E1:19_4_1_1_5_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A1</font>
 ;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_4_2"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">z11</font> = <font color="Maroon">z1</font>, <font color="Maroon">z22</font> = <font color="Maroon">z2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E5:19_4_1_1_5_4_2"/>
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z11</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z22</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E6:19_4_1_1_5_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_5_4"/><b>then </b>
<font color="Maroon">B1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_5_4"/>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:19_4_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="grsolv_1.html#E7:19"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">N</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon">y1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="grsolv_1.html#D1" target="_self">Def1</a>, , , <a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="grsolv_1.html#E4:19"><i><font color="Green">E38</font></i></a>, , , <a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E41</font></i></a>, <a class="ref" href="grsolv_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">N</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="grsolv_1.html#D1" target="_self">Def1</a>, <a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_4_1"/>
<font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Maroon">N</font> is <a href="group_1.html#V7">commutative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:19_4"/>
( <font color="Maroon">G2</font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">G1</font> &amp; ( for <font color="Olive">N</font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">G1</font>  st <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> holds <br/><font color="Maroon">G1</font> <a href="group_6.html#K6">./.</a> <font color="Olive">N</font> is <a href="group_1.html#V7">commutative</a> ) )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">z</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="grsolv_1.html#D1" target="_self">GRSOLV_1:def 1</a><br/>

<b>thus </b><a NAME="E14:19"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">h</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> &amp; <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> holds <br/>for <font color="Olive">G1</font>, <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of  <a href="group_6.html#K12">Image</a> <font color="Maroon">h</font>  st <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">G2</font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">G1</font> &amp; ( for <font color="Olive">N</font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">G1</font>  st <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Olive">G2</font> holds <br/><font color="Olive">G1</font> <a href="group_6.html#K6">./.</a> <font color="Olive">N</font> is <a href="group_1.html#V7">commutative</a> ) ) ) )
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div>
