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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>;<br/>

<a NAME="E1:9"/>
(  <a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>1</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> &amp;  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</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="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E2:9"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> is    <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>by </b><i><a class="ref" href="finseq_2.html#T15">FINSEQ_2:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</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="E4:9"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 ;<br/>
<a NAME="E5:9"/><i><font color="Green">E5</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E4:9"><i><font color="Green">E4</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E6:9"/>
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>2</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>2</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>2</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>2</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="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</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>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 ;<br/><a NAME="E2:9_1_1"/><i><font color="Green">E8</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:9"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:9_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:9"><i><font color="Green">E5</font></i></a>, <a class="txt" href="grsolv_1.html#E1:9_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:9_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E1:9_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="grsolv_1.html#E2:9_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:9_1_1"/>
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>4</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</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="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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>4</sub></font>;<br/>

<b>assume </b><a NAME="E1:9_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:9_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:9"><i><font color="Green">E5</font></i></a>, <a class="txt" href="grsolv_1.html#E1:9_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="grsolv_1.html#E3:9_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="grsolv_1.html#E4:9_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:9_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#R2">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T82">GROUP_6:82</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="group_1.html#V7">commutative</a>
 <b>by </b><i><a class="ref" href="group_6.html#T89">GROUP_6:89</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:9_1_1"/>
( <font color="Maroon">c<sub>5</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>4</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>4</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</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#E5:9"><i><font color="Green">E5</font></i></a>, <a class="txt" href="grsolv_1.html#E1:9_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="grsolv_1.html#E3:9_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="grsolv_1.html#E4:9_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">b<sub>2</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>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</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>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_6.html#K6">./.</a> <font color="Olive">b<sub>3</sub></font> is <a href="group_1.html#V7">commutative</a> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:9"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V1">strict</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="grsolv_1.html#V1" target="_self">solvable</a> )
 <b>by </b><i><a class="txt" href="grsolv_1.html#E5:9"><i><font color="Green">E5</font></i></a>, <a class="ref" href="grsolv_1.html#D1" target="_self">Def1</a></i>;<br/>


</div>
