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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <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>let </b><font color="Maroon" title="c2">H</font> be  <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>;<br/>



<b>consider </b><font color="Maroon" title="c3">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="E2:7"/><i><font color="Green" title="E4">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">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="c3">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="c3">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="c3">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="c3">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="c3">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="c3">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">I</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">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="Olive" title="b1">I</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> );<br/>
<a NAME="E3:7"/><i><font color="Green" title="E5">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="c3">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> <font color="Maroon" title="c2">H</font> 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="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:7_1"/>
<font color="Maroon" title="c4">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="c3">F</font></span>)</span>
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
<font color="Maroon" title="c4">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="c3">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="E3:7_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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/>
<b>then reconsider </b><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> as  <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>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">I</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> as  <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 <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="ref" href="group_2.html#T106" title="GROUP_2:th.106">GROUP_2:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">x</font> as    <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> <font color="Maroon" title="c2">H</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>take </b>
<font color="Maroon" title="c7">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">I</font>
;<br/>

<b>thus </b><a NAME="E7:7_1"/>
( <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">R</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="c2">H</font><b> such that </b><br/><a NAME="E5:7"/><i><font color="Green" title="E6">A3</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">R</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="c3">F</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#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="c3">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</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#E3:7"><i><font color="Green" title="E5">A2</font></i></a>);<br/></i>
<a NAME="E6:7"/><i><font color="Green" title="E7">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</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="c3">F</font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</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="E7:7"/><i><font color="Green" title="E8">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</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="E8:7"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c4">R</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="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<a NAME="E1:7_2"/>
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">R</font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E7:7"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E2:7_2"/><b>then </b>
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="c3">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:7"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">I</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="E4:7_2"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c4">R</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="c5">I</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:7_2"/>
<font color="Maroon" title="c4">R</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="c2">H</font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="group_2.html#T104" title="GROUP_2:th.104">GROUP_2:104</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:7"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c4">R</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">R</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="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>proof </b></a><div class="add">

<a NAME="E1:7_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</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="c3">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7"><i><font color="Green" title="E7">A4</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="c5">I</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="E3:7_3"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">R</font></span>)</span> &amp; <font color="Maroon" title="c4">R</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">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">R</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">R</font></span>)</span> = 
 <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="grsolv_1.html#E3:7_3"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="group_2.html#T103" title="GROUP_2:th.103">GROUP_2:103</a></i>
<br/>.= 
 <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="ref" href="group_2.html#T75" title="GROUP_2:th.75">GROUP_2:75</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:7"/><i><font color="Green" title="E11">A10</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="c4">R</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">R</font> holds <br/>for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</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 <font color="Maroon" title="c2">H</font>  st <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R</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">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R</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">H2</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">H1</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">H1</font>  st <font color="Olive" title="b4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H2</font> holds <br/><font color="Olive" title="b2">H1</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="7_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:7_4"/><i><font color="Green" title="E12">A11</font></i>: 
( <font color="Maroon" title="c5">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">R</font> &amp; <font color="Maroon" title="c5">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">R</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">H1</font>, <font color="Maroon" title="c7">H2</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 <font color="Maroon" title="c2">H</font>;<br/>

<b>assume </b><a NAME="E2:7_4"/><i><font color="Green" title="E13">A12</font></i>: 
( <font color="Maroon" title="c6">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">I</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="E4:7_4"/><i><font color="Green" title="E14">A13</font></i>: 
( <font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="grsolv_1.html#E1:7_4"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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><b> such that </b><br/><a NAME="E6:7_4"/><i><font color="Green" title="E15">A14</font></i>: 
( <font color="Maroon" title="c9">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">J</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="grsolv_1.html#E1:7_4"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<a NAME="E7:7_4"/><i><font color="Green" title="E16">A15</font></i>: 
( <font color="Maroon" title="c5">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="c3">F</font> &amp; <font color="Maroon" title="c5">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="c3">F</font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="grsolv_1.html#E1:7_4"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">J1</font> = <font color="Maroon" title="c9">J</font> as  <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="c8">I</font> <b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/>
<a NAME="E9:7_4"/><i><font color="Green" title="E17">A16</font></i>: 
( <font color="Maroon" title="c6">H1</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">I</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c7">H2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c10">J1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c2">H</font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7_4"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/>
<a NAME="E10:7_4"/>
for <font color="Olive" title="b1">N</font> being <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="c6">H1</font>  st <font color="Olive" title="b1">N</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> holds <br/><font color="Maroon" title="c6">H1</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="7_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">N</font> be  <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="c6">H1</font>;<br/>

<b>assume </b><a NAME="E1:7_4_1"/>
<font color="Maroon" title="c11">N</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">G3</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c8">I</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c10">J1</font><b> such that </b><br/><a NAME="E3:7_4_1"/><i><font color="Green" title="E18">A17</font></i>: 
<font color="Maroon" title="c6">H1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font>,<font color="Maroon" title="c12">G3</font> <a href="group_6.html#R1" title="GROUP_6:pred.1">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:7_4"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="grsolv_1.html#T4" target="_self" title="GRSOLV_1:th.4">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span>,<font color="Maroon" title="c12">G3</font><b> such that </b><br/><a NAME="E5:7_4_1"/><i><font color="Green" title="E19">A18</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:7_4_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="group_6.html#D15" title="GROUP_6:def.15">GROUP_6:def 15</a></i>;<br/>
<a NAME="E6:7_4_1"/><i><font color="Green" title="E20">A19</font></i>: 
( <font color="Maroon" title="c13">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  &amp; <font color="Maroon" title="c13">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a>  )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7_4_1"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a></i>;<br/>
<a NAME="E7:7_4_1"/>
<font color="Maroon" title="c8">I</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c10">J1</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#E2:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="grsolv_1.html#E7:7_4"><i><font color="Green" title="E16">A15</font></i></a></i>;<br/>
<a NAME="E8:7_4_1"/><b>then </b><i><font color="Green" title="E21">A20</font></i>: 
<font color="Maroon" title="c12">G3</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T62" title="GROUP_2:th.62">GROUP_2:62</a></i>;<br/>
<a NAME="E9:7_4_1"/>
<font color="Maroon" title="c6">H1</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">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="c6">H1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c16">a'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">G3</font><b> such that </b><br/><a NAME="E2:7_4_1_1_1"/><i><font color="Green" title="E22">A21</font></i>: 
<font color="Maroon" title="c16">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
 ;<br/><b>consider </b><font color="Maroon" title="c17">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">G3</font><b> such that </b><br/><a NAME="E4:7_4_1_1_1"/><i><font color="Green" title="E23">A22</font></i>: 
<font color="Maroon" title="c17">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font>
 ;<br/><a NAME="E5:7_4_1_1_1"/><i><font color="Green" title="E24">A23</font></i>: 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c12">G3</font> 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#E8:7_4_1"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="group_3.html#T2" title="GROUP_3:th.2">GROUP_3:2</a></i>;<br/><i><font color="Green" title="E25">A24</font></i>: <font color="Maroon" title="c16">a'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c17">b'</font> = 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c12">G3</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">a'</font>,<font color="Maroon" title="c17">b'</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/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c12">G3</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">b'</font>,<font color="Maroon" title="c16">a'</font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7_4_1_1_1"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c17">b'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c16">a'</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/><b>thus </b>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="c6">H1</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="c14">a</font>,<font color="Maroon" title="c15">b</font> = 
<font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</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"><font color="Maroon" title="c13">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:7_4_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="group_6.html#T64" title="GROUP_6:th.64">GROUP_6:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7_4_1_1_1"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4_1_1_1"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4_1_1_1"><i><font color="Green" title="E25">A24</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c15">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:7_4_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="group_6.html#T64" title="GROUP_6:th.64">GROUP_6:64</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="c6">H1</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="c15">b</font>,<font color="Maroon" title="c14">a</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>
<a NAME="E2:7_4_1_1"/><b>then </b>
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="c6">H1</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c11">N</font></span>)</span> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 
<b>by </b><i><a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:7_4_1_1"/>
<font color="Maroon" title="c6">H1</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_3.html#T2" title="GROUP_3:th.2">GROUP_3:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:7_4_1"/>
<font color="Maroon" title="c6">H1</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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:7_4"/>
( <font color="Maroon" title="c7">H2</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="c6">H1</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="c6">H1</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">H2</font> holds <br/><font color="Maroon" title="c6">H1</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#E9:7_4"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="grsolv_1.html#T1" target="_self" title="GRSOLV_1:th.1">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c4">R</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="E11:7"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c4">R</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="c2">H</font> &amp; <font color="Maroon" title="c4">R</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">R</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="c2">H</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">R</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">R</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="c2">H</font>  st <font color="Olive" title="b2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R</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">R</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#E2:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="grsolv_1.html#E8:7"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="grsolv_1.html#E9:7"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="grsolv_1.html#E10:7"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div>
