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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:19"/>
<font color="Maroon">c<sub>1</sub></font> is  <a href="grsolv_1.html#V1" target="_self">solvable</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<b>then consider </b><font color="Maroon">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>1</sub></font><b> such that </b><br/><a NAME="E3:19"/><i><font color="Green">E13</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 &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>1</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>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>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>1</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="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>4</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="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E4:19"/><i><font color="Green">E14</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>4</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> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> 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="19_1"><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="E1:19_1"/>
<font color="Maroon">c<sub>5</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>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</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/>
<a NAME="E4:19_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is  <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>then consider </b><font color="Maroon">c<sub>6</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="E6:19_1"/><i><font color="Green">E15</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>
 ;<br/>
<a NAME="E7:19_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>6</sub></font> is  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of  <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="grsolv_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E8:19_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E10:19_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>
<b>thus </b><a NAME="E11:19_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="grsolv_1.html#E10:19_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="group_3.html#K1">Subgroups</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:19"/><i><font color="Green">E15</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</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>4</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>4</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>5</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#E4:19"><i><font color="Green">E14</font></i></a>);<br/></i>
<a NAME="E7:19"/><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green">E17</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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#E6:19"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E18</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E7:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E10:19"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E11:19"/><b>then </b><i><font color="Green">E19</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:19"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</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>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</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="E2:19_3"/><i><font color="Green">E21</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> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="grsolv_1.html#E11:19"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:19_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E2:19_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="grsolv_1.html#T12" target="_self">Th12</a></i>;<br/>

<a NAME="E4:19_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>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E13</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>7</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="E6:19_3"/><i><font color="Green">E22</font></i>: 
( <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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</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>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>5</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>5</sub></font></span>)</span> = 
 <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E8:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="grsolv_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
 <a href="group_2.html#K5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T75">GROUP_2:75</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E13:19"/><i><font color="Green">E21</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>5</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>5</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  <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</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>5</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="19_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_4"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</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>5</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of  <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E2:19_4"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 ;<br/>

<b>then 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="E4:19_4"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>9</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>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</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="E6:19_4"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>10</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>6</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="grsolv_1.html#E2:19_4"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E7:19_4"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</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>9</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E9:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E8:19_4"/>
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>7</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>7</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="19_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>assume </b><a NAME="E1:19_4_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/><a NAME="E1:19_4_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E2:19_4_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 ;<br/><a NAME="E3:19_4_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4">Cosets</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4">Cosets</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="group_6.html#T22">GROUP_6:22</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E5:19_4_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E1:19_4_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="group_6.html#T28">GROUP_6:28</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E7:19_4_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E1:19_4_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="group_6.html#T28">GROUP_6:28</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="group_6.html#T17">GROUP_6:17</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="group_6.html#T17">GROUP_6:17</a></i>;<br/><a NAME="E10:19_4_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E11:19_4_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E13:19_4_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E15:19_4_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E11:19_4_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/><i><font color="Green">E35</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E13:19_4_1_1"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><i><font color="Green">E36</font></i>: <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E15:19_4_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><i><font color="Green">E37</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="grsolv_1.html#E18:19_4_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="grsolv_1.html#T14" target="_self">Th14</a></i>
;<br/><i><font color="Green">E38</font></i>: <font color="Maroon">c<sub>15</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="grsolv_1.html#E19:19_4_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="grsolv_1.html#T14" target="_self">Th14</a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E20:19_4_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E21:19_4_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E24:19_4_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="grsolv_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E25:19_4_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_5_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:19_4_1_1_5_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_1"><i><font color="Green">E41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:19_4_1_1_5_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_5_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_5"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_5_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_4_1_1_5_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</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>25</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E24:19_4_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:19_4_1_1_5_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_2"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:19_4_1_1_5_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4_1_1_5_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_5_2"><i><font color="Green">E44</font></i></a></i>;<br/><font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_5_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4_1_1_5_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
;<br/><b>hence </b><a NAME="E11:19_4_1_1_5_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_5_2"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_5"/>
<font color="Maroon">c<sub>22</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:19_4_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="grsolv_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E27:19_4_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_6_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:19_4_1_1_6_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_6_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_6_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E6:19_4_1_1_6_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_6_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_6_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_6_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_6"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_6_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_4_1_1_6_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</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>25</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E26:19_4_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_6_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:19_4_1_1_6_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_6_2"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E7:19_4_1_1_6_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4_1_1_6_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_6_2"><i><font color="Green">E46</font></i></a></i>;<br/><font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_6_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4_1_1_6_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
;<br/><b>hence </b><a NAME="E11:19_4_1_1_6_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_6_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_6"/>
<font color="Maroon">c<sub>23</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_6"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:19_4_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E9:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E3:19_4_1_1_7"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E9:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="grsolv_1.html#E1:19_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="grsolv_1.html#E4:19_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:19_4_1_1_7"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> holds the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>;<br/>

<a NAME="E1:19_4_1_1_7_1"/>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7"><i><font color="Green">E44</font></i></a>, <a class="ref" href="group_3.html#T2">GROUP_3:2</a></i>;<br/>
<b>hence </b><a NAME="E2:19_4_1_1_7_1"/>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_4_1_1_7"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>;<br/>
<a NAME="E6:19_4_1_1_7"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>24</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T27">GROUP_6:27</a></i>;<br/>
<a NAME="E7:19_4_1_1_7"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>24</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T27">GROUP_6:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4">Cosets</a> <font color="Maroon">c<sub>24</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E5:19_4_1_1_7"><i><font color="Green">E46</font></i></a>, <a class="ref" href="group_6.html#T16">GROUP_6:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <b>by </b><i><a class="txt" href="grsolv_1.html#E6:19_4_1_1_7"><i><font color="Green">E47</font></i></a>, <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>28</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <b>by </b><i><a class="txt" href="grsolv_1.html#E7:19_4_1_1_7"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E5:19_4_1_1_7"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E5:19_4_1_1_7"><i><font color="Green">E46</font></i></a></i>;<br/>
<i><font color="Green">E49</font></i>: <font color="Maroon">c<sub>31</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>32</sub></font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E4:19_4_1_1_7"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>32</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
;<br/>
<a NAME="E14:19_4_1_1_7"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>32</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>33</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_7_3_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:19_4_1_1_7_3_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><a NAME="E4:19_4_1_1_7_3_1"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_3_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/><a NAME="E6:19_4_1_1_7_3_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_7_3_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_3_1"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_7_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>31</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>33</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_7_3_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E3:19_4_1_1_7_3_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_3_2"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E5:19_4_1_1_7_3_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E6:19_4_1_1_7_3_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_3_2"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_7_3"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_7_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_7_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:19_4_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>31</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>33</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_7_4_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:19_4_1_1_7_4_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><a NAME="E4:19_4_1_1_7_4_1"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_4_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/><a NAME="E6:19_4_1_1_7_4_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E7:19_4_1_1_7_4_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_4_1"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:19_4_1_1_7_4"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>32</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_7_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>33</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4_1_1_7_4_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E3:19_4_1_1_7_4_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_4_2"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E5:19_4_1_1_7_4_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E6:19_4_1_1_7_4_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1_7_4_2"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:19_4_1_1_7_4"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>31</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_4_1_1_7_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E2:19_4_1_1_7_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:19_4_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E13:19_4_1_1_7"><i><font color="Green">E49</font></i></a>, <a class="txt" href="grsolv_1.html#E14:19_4_1_1_7"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>12</sub></font>,<font color="Maroon">c<sub>13</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"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="grsolv_1.html#E5:19_4_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="grsolv_1.html#E7:19_4_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="grsolv_1.html#E20:19_4_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="grsolv_1.html#E21:19_4_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="grsolv_1.html#E24:19_4_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="grsolv_1.html#E25:19_4_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="grsolv_1.html#E26:19_4_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="grsolv_1.html#E27:19_4_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="grsolv_1.html#E28:19_4_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K5">CosOp</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:19_4_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="group_6.html#D4">GROUP_6:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#T23">GROUP_6:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</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>
<b>hence </b><a NAME="E3:19_4_1"/>
<font color="Maroon">c<sub>7</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_1.html#D16">GROUP_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:19_4"/>
( <font color="Maroon">c<sub>8</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>7</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>7</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>7</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#E4:19_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="grsolv_1.html#E7:19_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="grsolv_1.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>5</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="E14:19"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K6">(Omega).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</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>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">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#K4">dom</a> <font color="Maroon">c<sub>5</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>5</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  <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</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>5</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#E3:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="grsolv_1.html#E6:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="grsolv_1.html#E12:19"><i><font color="Green">E20</font></i></a>, <a class="txt" href="grsolv_1.html#E13:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div>
