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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="grsolv_1.html#V1" target="_self">solvable</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:7"/><i><font color="Green">E4</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="group_6.html#K6">./.</a> <font color="Olive">b<sub>4</sub></font> is <a href="group_1.html#V7">commutative</a> ) ) ) )
 <b>by </b><i><a class="ref" href="grsolv_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> );<br/>
<a NAME="E3:7"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:7_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="group_2.html#T106">GROUP_2:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E7:7_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:7"/><i><font color="Green">E6</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="grsolv_1.html#E3:7"><i><font color="Green">E5</font></i></a>);<br/></i>
<a NAME="E6:7"/><i><font color="Green">E7</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:7"/><i><font color="Green">E8</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:7"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>2</sub></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">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E7:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E2:7_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:7_2"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="group_2.html#T104">GROUP_2:104</a></i>
<br/>.= 
 <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>2</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E9:7"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>2</sub></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">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:7_3"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="grsolv_1.html#E3:7_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="group_2.html#T103">GROUP_2:103</a></i>
<br/>.= 
 <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T75">GROUP_2:75</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:7"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="group_6.html#K6">./.</a> <font color="Olive">b<sub>4</sub></font> is <a href="group_1.html#V7">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">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:7_4"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:7_4"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:7_4"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="grsolv_1.html#E1:7_4"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:7_4"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="grsolv_1.html#E1:7_4"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E7:7_4"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="grsolv_1.html#E1:7_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green">E14</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E9:7_4"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7_4"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green">E14</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E10:7_4"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Olive">b<sub>1</sub></font> is <a href="group_1.html#V7">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">c<sub>11</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>

<b>assume </b><a NAME="E1:7_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>8</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E3:7_4_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="group_6.html#R1">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:7_4"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green">E14</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green">E15</font></i></a>, <a class="ref" href="grsolv_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E5:7_4_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:7_4_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="group_6.html#D15">GROUP_6:def 15</a></i>;<br/>
<a NAME="E6:7_4_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="group_6.html#NR1" target="_self">is_monomorphism</a>  )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7_4_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="group_6.html#D14">GROUP_6:def 14</a></i>;<br/>
<a NAME="E7:7_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="group_1.html#V7">commutative</a>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4"><i><font color="Green">E14</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="grsolv_1.html#E7:7_4"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E8:7_4_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="group_1.html#V7">commutative</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T62">GROUP_2:62</a></i>;<br/>
<a NAME="E9:7_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="group_1.html#V7">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">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E2:7_4_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E4:7_4_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E5:7_4_1_1_1"/><i><font color="Green">E24</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>12</sub></font> is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E8:7_4_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_3.html#T2">GROUP_3:2</a></i>;<br/><i><font color="Green">E25</font></i>: <font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:7_4_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:7_4_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="group_6.html#T64">GROUP_6:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></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">E22</font></i></a>, <a class="txt" href="grsolv_1.html#E4:7_4_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="grsolv_1.html#E6:7_4_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:7_4_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="group_6.html#T64">GROUP_6:64</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E2:7_4_1_1"/><b>then </b>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:7_4_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="group_1.html#V7">commutative</a>
 <b>by </b><i><a class="ref" href="group_3.html#T2">GROUP_3:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:7_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="group_1.html#V7">commutative</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:7_4"/>
( <font color="Maroon">c<sub>7</sub></font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#K6">./.</a> <font color="Olive">b<sub>1</sub></font> is <a href="group_1.html#V7">commutative</a> ) )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E9:7_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="grsolv_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="grsolv_1.html#D1" target="_self">GRSOLV_1:def 1</a><br/>

<b>thus </b><a NAME="E11:7"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="group_6.html#K6">./.</a> <font color="Olive">b<sub>4</sub></font> is <a href="group_1.html#V7">commutative</a> ) ) ) )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E5:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="grsolv_1.html#E8:7"><i><font color="Green">E9</font></i></a>, <a class="txt" href="grsolv_1.html#E9:7"><i><font color="Green">E10</font></i></a>, <a class="txt" href="grsolv_1.html#E10:7"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


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