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

<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">h</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>;<br/>



<b>assume </b><a NAME="E1:19"/>
<font color="Maroon" title="c1">G</font> is  <a href="grsolv_1.html#V1" title="GRSOLV_1:attr.1">solvable</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:19"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/>for <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">G2</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">G2</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">G1</font> &amp; ( for <font color="Olive" title="b4">N</font> being <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">G1</font>  st <font color="Olive" title="b4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b2">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Olive" title="b4">N</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> ) ) ) )
 <b>by </b><i><a class="ref" href="grsolv_1.html#D1" target="_self" title="GRSOLV_1:def.1">Def1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">J</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Olive" title="b1">J</font> );<br/>
<a NAME="E4:19"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">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" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E4:19_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">A</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:19_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 ;<br/>
<a NAME="E7:19_1"/>
<font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font>
 
<b>by </b><i><a class="ref" href="grsolv_1.html#T11" target="_self" title="GRSOLV_1:th.11">Th11</a></i>;<br/>
<a NAME="E8:19_1"/><b>then </b>
<font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span><b> such that </b><br/><a NAME="E10:19_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c6">A</font>
 ;<br/>
<b>thus </b><a NAME="E11:19_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="grsolv_1.html#E10:19_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">z</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span><b> such that </b><br/><a NAME="E6:19"/><i><font color="Green" title="E15">A5</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="grsolv_1.html#E4:19"><i><font color="Green" title="E14">A2</font></i></a>);<br/></i>
<a NAME="E7:19"/><i><font color="Green" title="E16">A6</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E17">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E18">A8</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E7:19"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E10:19"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E11:19"/><b>then </b><i><font color="Green" title="E19">A9</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:19"/><i><font color="Green" title="E20">A10</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span> &amp; <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">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" title="c6">A</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:19_3"/><i><font color="Green" title="E21">A11</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c6">A</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="grsolv_1.html#E11:19"><i><font color="Green" title="E19">A9</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:19_3"/>
<font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E2:19_3"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="grsolv_1.html#T12" target="_self" title="GRSOLV_1:th.12">Th12</a></i>;<br/>

<a NAME="E4:19_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">B</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:19_3"/><i><font color="Green" title="E22">A12</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> &amp; <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
 <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c2">H</font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_3"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="grsolv_1.html#T12" target="_self" title="GRSOLV_1:th.12">Th12</a></i>
<br/>.= 
 <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T75" title="GROUP_2:th.75">GROUP_2:75</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E13:19"/><i><font color="Green" title="E21">A13</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font> holds <br/>for <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">G2</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font>  st <font color="Olive" title="b2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">G2</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">G1</font> &amp; ( for <font color="Olive" title="b4">N</font> being <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">G1</font>  st <font color="Olive" title="b4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b2">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Olive" title="b4">N</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">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" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19_4"/><i><font color="Green" title="E22">A14</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">G1</font>, <font color="Maroon" title="c8">G2</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font>;<br/>

<b>assume </b><a NAME="E2:19_4"/><i><font color="Green" title="E23">A15</font></i>: 
( <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:19_4"/><i><font color="Green" title="E24">A16</font></i>: 
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c7">G1</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green" title="E22">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B</font> being  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:19_4"/><i><font color="Green" title="E25">A17</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">G2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="grsolv_1.html#E2:19_4"><i><font color="Green" title="E23">A15</font></i></a></i>;<br/>
<a NAME="E7:19_4"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c10">B</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c9">A</font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E9:19"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green" title="E25">A17</font></i></a></i>;<br/>
<a NAME="E8:19_4"/>
for <font color="Olive" title="b1">N</font> being <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c7">G1</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G2</font> holds <br/><font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Olive" title="b1">N</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">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" title="c11">N</font> be  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c7">G1</font>;<br/>

<b>assume </b><a NAME="E1:19_4_1"/><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span>;<br/><a NAME="E1:19_4_1_1"/><i><font color="Green" title="E28">A20</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c13">y</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E2:19_4_1_1"/>
( <font color="Maroon" title="c12">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="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span> &amp; <font color="Maroon" title="c13">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="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span> )
 ;<br/><a NAME="E3:19_4_1_1"/><b>then </b><i><font color="Green" title="E29">A21</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#K4" title="GROUP_6:func.4">Cosets</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#K4" title="GROUP_6:func.4">Cosets</a> <font color="Maroon" title="c11">N</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T22" title="GROUP_6:th.22">GROUP_6:22</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font><b> such that </b><br/><a NAME="E5:19_4_1_1"/><i><font color="Green" title="E30">A22</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c14">a</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E1:19_4_1_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="group_6.html#T28" title="GROUP_6:th.28">GROUP_6:28</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font><b> such that </b><br/><a NAME="E7:19_4_1_1"/><i><font color="Green" title="E31">A23</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c15">b</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E1:19_4_1_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="group_6.html#T28" title="GROUP_6:th.28">GROUP_6:28</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c12">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="group_6.html#T17" title="GROUP_6:th.17">GROUP_6:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c13">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="group_6.html#T17" title="GROUP_6:th.17">GROUP_6:17</a></i>;<br/><a NAME="E10:19_4_1_1"/>
( <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c15">b</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E11:19_4_1_1"/><b>then </b><i><font color="Green" title="E32">A24</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K1" title="GRSOLV_1:func.1">|</a> <font color="Maroon" title="c9">A</font></span>)</span> &amp; <font color="Maroon" title="c15">b</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K1" title="GRSOLV_1:func.1">|</a> <font color="Maroon" title="c9">A</font></span>)</span> )
 ;<br/><b>then consider </b><font color="Maroon" title="c18">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font><b> such that </b><br/><a NAME="E13:19_4_1_1"/><i><font color="Green" title="E33">A25</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K1" title="GRSOLV_1:func.1">|</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a1</font>
 <b>by </b><i><a class="ref" href="group_6.html#T54" title="GROUP_6:th.54">GROUP_6:54</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font><b> such that </b><br/><a NAME="E15:19_4_1_1"/><i><font color="Green" title="E34">A26</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K1" title="GRSOLV_1:func.1">|</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E11:19_4_1_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="group_6.html#T54" title="GROUP_6:th.54">GROUP_6:54</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">a1</font> = <font color="Maroon" title="c18">a1</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/><b>reconsider </b><font color="Maroon" title="c21">b1</font> = <font color="Maroon" title="c19">b1</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="E18:19_4_1_1"/><i><font color="Green" title="E35">A27</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">a1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E13:19_4_1_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E19:19_4_1_1"/><i><font color="Green" title="E36">A28</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">b1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E15:19_4_1_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><i><font color="Green" title="E37">A29</font></i>: <font color="Maroon" title="c14">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c11">N</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">a1</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="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c10">B</font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="grsolv_1.html#E18:19_4_1_1"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="group_6.html#T2" title="GROUP_6:th.2">GROUP_6:2</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
<b>by </b><i><a class="ref" href="grsolv_1.html#T14" target="_self" title="GRSOLV_1:th.14">Th14</a></i>
;<br/><i><font color="Green" title="E38">A30</font></i>: <font color="Maroon" title="c15">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c11">N</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">b1</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="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c10">B</font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="grsolv_1.html#E19:19_4_1_1"><i><font color="Green" title="E36">A28</font></i></a>, <a class="ref" href="group_6.html#T2" title="GROUP_6:th.2">GROUP_6:2</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
<b>by </b><i><a class="ref" href="grsolv_1.html#T14" target="_self" title="GRSOLV_1:th.14">Th14</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c22">x2</font> = <font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E20:19_4_1_1"><i><font color="Green" title="E37">A29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">y2</font> = <font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E21:19_4_1_1"><i><font color="Green" title="E38">A30</font></i></a></i>;<br/><a NAME="E24:19_4_1_1"/><i><font color="Green" title="E39">A31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="grsolv_1.html#T15" target="_self" title="GRSOLV_1:th.15">Th15</a></i>;<br/><a NAME="E25:19_4_1_1"/><i><font color="Green" title="E40">A32</font></i>: 
<font color="Maroon" title="c22">x2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c23">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">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" title="c24">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_3_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">x2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c23">y2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c25">z1</font>, <font color="Maroon" title="c26">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font><b> such that </b><br/><a NAME="E3:19_4_1_1_3_1"/><i><font color="Green" title="E41">A33</font></i>: 
( <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c26">z2</font> &amp; <font color="Maroon" title="c25">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">x2</font> &amp; <font color="Maroon" title="c26">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y2</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c27">z11</font> = <font color="Maroon" title="c25">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_3_1"><i><font color="Green" title="E41">A33</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">z22</font> = <font color="Maroon" title="c26">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_3_1"><i><font color="Green" title="E41">A33</font></i></a></i>;<br/><a NAME="E6:19_4_1_1_3_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c28">z22</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_3_1"><i><font color="Green" title="E41">A33</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_3_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_3_1"><i><font color="Green" title="E41">A33</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" title="E41">A34</font></i>: 
<font color="Maroon" title="c22">x2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c23">y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c24">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_3_2"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c25">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:19_4_1_1_3_2"/><i><font color="Green" title="E42">A35</font></i>: 
( <font color="Maroon" title="c25">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 <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> &amp; <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E24:19_4_1_1"><i><font color="Green" title="E39">A31</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c25">x</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="txt" href="grsolv_1.html#E3:19_4_1_1_3_2"><i><font color="Green" title="E42">A35</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c27">z1</font>, <font color="Maroon" title="c28">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:19_4_1_1_3_2"/><i><font color="Green" title="E43">A36</font></i>: 
( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c28">z2</font> &amp; <font color="Maroon" title="c27">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c28">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_3_2"><i><font color="Green" title="E42">A35</font></i></a></i>;<br/><a NAME="E7:19_4_1_1_3_2"/><i><font color="Green" title="E44">A37</font></i>: 
( <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">x2</font> &amp; <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y2</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4_1_1_3_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">z11</font> = <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c30">z22</font> = <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_3_2"><i><font color="Green" title="E44">A37</font></i></a></i>;<br/><font color="Maroon" title="c24">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">z1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">z2</font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_3_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4_1_1_3_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c29">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c30">z22</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="E11:19_4_1_1_3_2"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">x2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c23">y2</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_3_2"><i><font color="Green" title="E44">A37</font></i></a></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" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">x2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c23">y2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_3"/>
<font color="Maroon" title="c22">x2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c23">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_3"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:19_4_1_1"/><i><font color="Green" title="E41">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="grsolv_1.html#T15" target="_self" title="GRSOLV_1:th.15">Th15</a></i>;<br/><a NAME="E27:19_4_1_1"/><i><font color="Green" title="E42">A39</font></i>: 
<font color="Maroon" title="c23">y2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c22">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">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" title="c24">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_4_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c22">x2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c25">z1</font>, <font color="Maroon" title="c26">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font><b> such that </b><br/><a NAME="E3:19_4_1_1_4_1"/><i><font color="Green" title="E43">A40</font></i>: 
( <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c26">z2</font> &amp; <font color="Maroon" title="c25">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y2</font> &amp; <font color="Maroon" title="c26">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">x2</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c27">z11</font> = <font color="Maroon" title="c25">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_4_1"><i><font color="Green" title="E43">A40</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">z22</font> = <font color="Maroon" title="c26">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_4_1"><i><font color="Green" title="E43">A40</font></i></a></i>;<br/><a NAME="E6:19_4_1_1_4_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c28">z22</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_4_1"><i><font color="Green" title="E43">A40</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_4_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_4_1"><i><font color="Green" title="E43">A40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_4"/><b>then </b><i><font color="Green" title="E43">A41</font></i>: 
<font color="Maroon" title="c23">y2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c22">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c24">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_4_2"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c25">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:19_4_1_1_4_2"/><i><font color="Green" title="E44">A42</font></i>: 
( <font color="Maroon" title="c25">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 <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> &amp; <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E26:19_4_1_1"><i><font color="Green" title="E41">A38</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c25">x</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="txt" href="grsolv_1.html#E3:19_4_1_1_4_2"><i><font color="Green" title="E44">A42</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c27">z1</font>, <font color="Maroon" title="c28">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:19_4_1_1_4_2"/><i><font color="Green" title="E45">A43</font></i>: 
( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c28">z2</font> &amp; <font color="Maroon" title="c27">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c28">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_4_2"><i><font color="Green" title="E44">A42</font></i></a></i>;<br/><a NAME="E7:19_4_1_1_4_2"/><i><font color="Green" title="E46">A44</font></i>: 
( <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y2</font> &amp; <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">x2</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4_1_1_4_2"><i><font color="Green" title="E45">A43</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">z11</font> = <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c30">z22</font> = <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">G1</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_4_2"><i><font color="Green" title="E46">A44</font></i></a></i>;<br/><font color="Maroon" title="c24">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">z1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">z2</font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_4_2"><i><font color="Green" title="E44">A42</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4_1_1_4_2"><i><font color="Green" title="E45">A43</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c29">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c30">z22</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="E11:19_4_1_1_4_2"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c22">x2</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_4_2"><i><font color="Green" title="E46">A44</font></i></a></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" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">y2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c22">x2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_4"/>
<font color="Maroon" title="c23">y2</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c22">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></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">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_4"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:19_4_1_1"/><i><font color="Green" title="E43">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">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" title="c24">K</font> = <font color="Maroon" title="c10">B</font> as  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c9">A</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E9:19"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green" title="E25">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">a2</font> = <font color="Maroon" title="c20">a1</font>, <font color="Maroon" title="c26">b2</font> = <font color="Maroon" title="c21">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font> ;<br/>
<a NAME="E3:19_4_1_1_5"/><i><font color="Green" title="E44">A46</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font> is  <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E9:19"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green" title="E25">A17</font></i></a></i>;<br/>
<a NAME="E4:19_4_1_1_5"/><i><font color="Green" title="E45">A47</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> holds  the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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" title="c27">a</font>, <font color="Maroon" title="c28">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span>;<br/>

<a NAME="E1:19_4_1_1_5_1"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5"><i><font color="Green" title="E44">A46</font></i></a>, <a class="ref" href="group_3.html#T2" title="GROUP_3:th.2">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" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c27">a</font>,<font color="Maroon" title="c28">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c28">b</font>,<font color="Maroon" title="c27">a</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_4_1_1_5"/><i><font color="Green" title="E46">A48</font></i>: 
( <font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">a2</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c24">K</font> &amp; <font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">b2</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c24">K</font> )
 
<b>by </b><i><a class="ref" href="group_6.html#T2" title="GROUP_6:th.2">GROUP_6:2</a></i>;<br/>
<a NAME="E6:19_4_1_1_5"/><i><font color="Green" title="E47">A49</font></i>: 
<font color="Maroon" title="c25">a2</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c24">K</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T27" title="GROUP_6:th.27">GROUP_6:27</a></i>;<br/>
<a NAME="E7:19_4_1_1_5"/><i><font color="Green" title="E48">A50</font></i>: 
<font color="Maroon" title="c26">b2</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c24">K</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T27" title="GROUP_6:th.27">GROUP_6:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">a12</font> = <font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font>, <font color="Maroon" title="c28">b12</font> = <font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#K4" title="GROUP_6:func.4">Cosets</a> <font color="Maroon" title="c24">K</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E5:19_4_1_1_5"><i><font color="Green" title="E46">A48</font></i></a>, <a class="ref" href="group_6.html#T16" title="GROUP_6:th.16">GROUP_6:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">a11</font> = <font color="Maroon" title="c27">a12</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4_1_1_5"><i><font color="Green" title="E47">A49</font></i></a>, <a class="ref" href="group_6.html#T2" title="GROUP_6:th.2">GROUP_6:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">b11</font> = <font color="Maroon" title="c28">b12</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_5"><i><font color="Green" title="E48">A50</font></i></a>, <a class="ref" href="group_6.html#T2" title="GROUP_6:th.2">GROUP_6:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">A1</font> = <font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">A</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E5:19_4_1_1_5"><i><font color="Green" title="E46">A48</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">B1</font> = <font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">A</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E5:19_4_1_1_5"><i><font color="Green" title="E46">A48</font></i></a></i>;<br/>
<i><font color="Green" title="E49">A51</font></i>: <font color="Maroon" title="c31">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c32">B1</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K5" title="GROUP_6:func.5">CosOp</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c29">a11</font>,<font color="Maroon" title="c30">b11</font>
<b>by </b><i><a class="ref" href="group_6.html#D4" title="GROUP_6:def.4">GROUP_6:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c27">a12</font>,<font color="Maroon" title="c28">b12</font>
<b>by </b><i><a class="ref" href="group_6.html#T23" title="GROUP_6:th.23">GROUP_6:23</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c30">b11</font>,<font color="Maroon" title="c29">a11</font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E4:19_4_1_1_5"><i><font color="Green" title="E45">A47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5" title="GROUP_6:func.5">CosOp</a> <font color="Maroon" title="c24">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c28">b12</font>,<font color="Maroon" title="c27">a12</font>
<b>by </b><i><a class="ref" href="group_6.html#T23" title="GROUP_6:th.23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon" title="c32">B1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c31">A1</font>
<b>by </b><i><a class="ref" href="group_6.html#D4" title="GROUP_6:def.4">GROUP_6:def 4</a></i>
;<br/>
<a NAME="E14:19_4_1_1_5"/><i><font color="Green" title="E50">A52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c32">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" title="c33">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_5_3_1"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c35">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_3_1"/><i><font color="Green" title="E51">A53</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> &amp; <font color="Maroon" title="c34">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> )
 ;<br/><a NAME="E4:19_4_1_1_5_3_1"/>
( <font color="Maroon" title="c34">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">A1</font> &amp; <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">B1</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_3_1"><i><font color="Green" title="E51">A53</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">z11</font> = <font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c37">z22</font> = <font color="Maroon" title="c35">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font> ;<br/><a NAME="E6:19_4_1_1_5_3_1"/>
<font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c37">z22</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="E7:19_4_1_1_5_3_1"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c32">B1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_3_1"><i><font color="Green" title="E51">A53</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" title="E51">A54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c32">B1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c33">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_5_3_2"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c32">B1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c35">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_3_2"/><i><font color="Green" title="E52">A55</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> &amp; <font color="Maroon" title="c34">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">A1</font> &amp; <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">B1</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c36">z11</font> = <font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c37">z22</font> = <font color="Maroon" title="c35">z2</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="txt" href="grsolv_1.html#E3:19_4_1_1_5_3_2"><i><font color="Green" title="E52">A55</font></i></a></i>;<br/><a NAME="E5:19_4_1_1_5_3_2"/>
<font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c37">z22</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="E6:19_4_1_1_5_3_2"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_3_2"><i><font color="Green" title="E52">A55</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_5_3"/><b>then </b>
<font color="Maroon" title="c31">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c32">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c32">B1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_5_3"><i><font color="Green" title="E51">A54</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">B1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c31">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" title="c33">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_5_4_1"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c35">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_4_1"/><i><font color="Green" title="E51">A56</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> &amp; <font color="Maroon" title="c34">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font> )
 ;<br/><a NAME="E4:19_4_1_1_5_4_1"/>
( <font color="Maroon" title="c34">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">B1</font> &amp; <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">A1</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_4_1"><i><font color="Green" title="E51">A56</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">z11</font> = <font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c37">z22</font> = <font color="Maroon" title="c35">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font> ;<br/><a NAME="E6:19_4_1_1_5_4_1"/>
<font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c37">z22</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="E7:19_4_1_1_5_4_1"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">B1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c31">A1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_4_1"><i><font color="Green" title="E51">A56</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" title="E51">A57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c32">B1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c31">A1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c33">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_5_4_2"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">B1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c31">A1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c35">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_4_2"/><i><font color="Green" title="E52">A58</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> &amp; <font color="Maroon" title="c34">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">B1</font> &amp; <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">A1</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c36">z11</font> = <font color="Maroon" title="c34">z1</font>, <font color="Maroon" title="c37">z22</font> = <font color="Maroon" title="c35">z2</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="txt" href="grsolv_1.html#E3:19_4_1_1_5_4_2"><i><font color="Green" title="E52">A58</font></i></a></i>;<br/><a NAME="E5:19_4_1_1_5_4_2"/>
<font color="Maroon" title="c34">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">z11</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c37">z22</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="E6:19_4_1_1_5_4_2"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_4_2"><i><font color="Green" title="E52">A58</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_5_4"/><b>then </b>
<font color="Maroon" title="c32">B1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c31">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">B1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c31">A1</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_5_4"><i><font color="Green" title="E51">A57</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</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="c20">a1</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c10">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E13:19_4_1_1_5"><i><font color="Green" title="E49">A51</font></i></a>, <a class="txt" href="grsolv_1.html#E14:19_4_1_1_5"><i><font color="Green" title="E50">A52</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon" title="c12">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">y</font> = 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5" title="GROUP_6:func.5">CosOp</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="ref" href="group_6.html#T23" title="GROUP_6:th.23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon" title="c17">y1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c16">x1</font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="grsolv_1.html#E5:19_4_1_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="grsolv_1.html#E7:19_4_1_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="grsolv_1.html#E20:19_4_1_1"><i><font color="Green" title="E37">A29</font></i></a>, <a class="txt" href="grsolv_1.html#E21:19_4_1_1"><i><font color="Green" title="E38">A30</font></i></a>, <a class="txt" href="grsolv_1.html#E24:19_4_1_1"><i><font color="Green" title="E39">A31</font></i></a>, <a class="txt" href="grsolv_1.html#E25:19_4_1_1"><i><font color="Green" title="E40">A32</font></i></a>, <a class="txt" href="grsolv_1.html#E26:19_4_1_1"><i><font color="Green" title="E41">A38</font></i></a>, <a class="txt" href="grsolv_1.html#E27:19_4_1_1"><i><font color="Green" title="E42">A39</font></i></a>, <a class="txt" href="grsolv_1.html#E28:19_4_1_1"><i><font color="Green" title="E43">A45</font></i></a>, <a class="ref" href="group_6.html#D4" title="GROUP_6:def.4">GROUP_6:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5" title="GROUP_6:func.5">CosOp</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="group_6.html#D4" title="GROUP_6:def.4">GROUP_6:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="ref" href="group_6.html#T23" title="GROUP_6:th.23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon" title="c13">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_4_1"/>
<font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:19_4"/>
( <font color="Maroon" title="c8">G2</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c7">G1</font> &amp; ( for <font color="Olive" title="b1">N</font> being <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c7">G1</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G2</font> holds <br/><font color="Maroon" title="c7">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Olive" title="b1">N</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> ) )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="grsolv_1.html#E7:19_4"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="grsolv_1.html#T16" target="_self" title="GRSOLV_1:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c5">z</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="grsolv_1.html#D1" target="_self" title="GRSOLV_1:def.1">GRSOLV_1:def 1</a><br/>

<b>thus </b><a NAME="E14:19"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span> &amp; <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">z</font> holds <br/>for <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">G2</font> being <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font>  st <font color="Olive" title="b2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">G2</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">G1</font> &amp; ( for <font color="Olive" title="b4">N</font> being <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">G1</font>  st <font color="Olive" title="b4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G2</font> holds <br/><font color="Olive" title="b2">G1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Olive" title="b4">N</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> ) ) ) )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="grsolv_1.html#E12:19"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="grsolv_1.html#E13:19"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div>
