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

<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c3">s1</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c4">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>







<b>assume </b><a NAME="E1:180"/><i><font color="Green" title="E153">A1</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:180"/><i><font color="Green" title="E154">A2</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> holds <br/><font color="Olive" title="b1">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">f1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font>;<br/>
<a NAME="E3:180"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
;<br/>
<a NAME="E4:180"/><b>then </b><i><font color="Green" title="E155">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:180_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_9.html#E4:180"><i><font color="Green" title="E155">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_1"><b>suppose </b></a><a NAME="E1:180_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:180_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:180"><i><font color="Green" title="E153">A1</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_2"><b>suppose </b></a><a NAME="E1:180_1_2"/><i><font color="Green" title="E156">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><div><i><font color="Green" title="E157">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:180_1_2_1"/>
not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><a NAME="E2:180_1_2_1"/><b>then </b>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>hence </b><a NAME="E3:180_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E1:180_1_2"><i><font color="Green" title="E156">A4</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E3:180_1_2"/><b>then </b><i><font color="Green" title="E158">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E4:180_1_2"/><b>then </b><i><font color="Green" title="E159">A7</font></i>: 
( <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:180_1_2"><i><font color="Green" title="E156">A4</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/><a NAME="E5:180_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:180"><i><font color="Green" title="E153">A1</font></i></a>, <a class="txt" href="group_9.html#E1:180_1_2"><i><font color="Green" title="E156">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:180_1_2"/><b>then </b><i><font color="Green" title="E160">A8</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:180_1_2"/><i><font color="Green" title="E161">A9</font></i>: 
for <font color="Olive" title="b1">H1</font> being   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">N1</font> being <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H1</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b2">N1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:180_1_2"><i><font color="Green" title="E157">A5</font></i></a>, <a class="txt" href="group_9.html#E1:180"><i><font color="Green" title="E153">A1</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>set </b><font color="Maroon" title="c6">H1</font> =  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">N1</font> =  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T16" target="_self" title="GROUP_9:th.16">Th16</a></i>;<br/><a NAME="E9:180_1_2"/><i><font color="Green" title="E162">A10</font></i>: 
 <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_9.html#T15" target="_self" title="GROUP_9:th.15">Th15</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">N1</font> = <font color="Maroon" title="c7">N1</font> as  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font> ;<br/><a NAME="E11:180_1_2"/>
<span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:180"><i><font color="Green" title="E153">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">H</font> = <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> <b>by </b><i><a class="ref" href="group_9.html#T102" target="_self" title="GROUP_9:th.102">Th102</a></i>;<br/><a NAME="E13:180_1_2"/>
<font color="Maroon" title="c9">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E2:180"><i><font color="Green" title="E154">A2</font></i></a></i>;<br/><a NAME="E14:180_1_2"/><b>then </b><i><font color="Green" title="E163">A11</font></i>: 
<span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c8">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E4:180_1_2"><i><font color="Green" title="E159">A7</font></i></a>, <a class="txt" href="group_9.html#E7:180_1_2"><i><font color="Green" title="E161">A9</font></i></a>, <a class="txt" href="group_9.html#E6:180_1_2"><i><font color="Green" title="E160">A8</font></i></a></i>;<br/><a NAME="E15:180_1_2"/>
 <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c8">N1</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E9:180_1_2"><i><font color="Green" title="E162">A10</font></i></a>, <a class="ref" href="group_9.html#T56" target="_self" title="GROUP_9:th.56">Th56</a></i>;<br/><a NAME="E16:180_1_2"/><b>then </b><i><font color="Green" title="E164">A12</font></i>: 
 <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E14:180_1_2"><i><font color="Green" title="E163">A11</font></i></a>, <a class="ref" href="group_9.html#T58" target="_self" title="GROUP_9:th.58">Th58</a></i>;<br/><a NAME="E17:180_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="txt" href="group_9.html#E6:180_1_2"><i><font color="Green" title="E160">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E18:180_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:180_1_2"><i><font color="Green" title="E156">A4</font></i></a>, <a class="txt" href="group_9.html#E3:180_1_2"><i><font color="Green" title="E158">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:180_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="txt" href="group_9.html#E6:180_1_2"><i><font color="Green" title="E160">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E20:180_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:180_1_2"><i><font color="Green" title="E156">A4</font></i></a>, <a class="txt" href="group_9.html#E3:180_1_2"><i><font color="Green" title="E158">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><a NAME="E21:180_1_2"/>
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E16:180_1_2"><i><font color="Green" title="E164">A12</font></i></a>, <a class="txt" href="group_9.html#E9:180_1_2"><i><font color="Green" title="E162">A10</font></i></a>, <a class="txt" href="group_9.html#E4:180_1_2"><i><font color="Green" title="E159">A7</font></i></a>, <a class="txt" href="group_9.html#E6:180_1_2"><i><font color="Green" title="E160">A8</font></i></a>, <a class="ref" href="group_9.html#T42" target="_self" title="GROUP_9:th.42">Th42</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_3"><b>suppose </b></a><a NAME="E1:180_1_3"/><i><font color="Green" title="E156">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><div><i><font color="Green" title="E157">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:180_1_3_1"/>
not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><a NAME="E2:180_1_3_1"/><b>then </b>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>hence </b><a NAME="E3:180_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E1:180_1_3"><i><font color="Green" title="E156">A13</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E3:180_1_3"/><b>then </b><i><font color="Green" title="E158">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E4:180_1_3"/><i><font color="Green" title="E159">A16</font></i>: 
0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:180_1_3"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="group_9.html#E1:180"><i><font color="Green" title="E153">A1</font></i></a></i>;<br/><a NAME="E6:180_1_3"/><b>then </b><i><font color="Green" title="E160">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:180_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:180_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="group_9.html#E4:180_1_3"><i><font color="Green" title="E159">A16</font></i></a>, <a class="txt" href="group_9.html#E3:180_1_3"><i><font color="Green" title="E158">A15</font></i></a></i>;<br/><a NAME="E9:180_1_3"/><b>then </b><i><font color="Green" title="E161">A18</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:180_1_3"/><b>then </b><i><font color="Green" title="E162">A19</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:180_1_3"/><b>then </b>
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">H1</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></i>;<br/><a NAME="E13:180_1_3"/>
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="group_9.html#E6:180_1_3"><i><font color="Green" title="E160">A17</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:180_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="group_9.html#E2:180_1_3"><i><font color="Green" title="E157">A14</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E15:180_1_3"/><b>then </b><i><font color="Green" title="E163">A20</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:180_1_3"/><b>then </b><i><font color="Green" title="E164">A21</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:180_1_3"/><b>then </b>
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">N1</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">N1</font> = <font color="Maroon" title="c7">N1</font> as  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E10:180_1_3"><i><font color="Green" title="E162">A19</font></i></a>, <a class="txt" href="group_9.html#E16:180_1_3"><i><font color="Green" title="E164">A21</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/><a NAME="E20:180_1_3"/>
<span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:180"><i><font color="Green" title="E153">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">H</font> = <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> <b>by </b><i><a class="ref" href="group_9.html#T102" target="_self" title="GROUP_9:th.102">Th102</a></i>;<br/><a NAME="E22:180_1_3"/>
<font color="Maroon" title="c9">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E2:180"><i><font color="Green" title="E154">A2</font></i></a></i>;<br/><a NAME="E23:180_1_3"/><b>then </b><i><font color="Green" title="E165">A22</font></i>: 
<font color="Maroon" title="c6">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c8">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E2:180_1_3"><i><font color="Green" title="E157">A14</font></i></a>, <a class="txt" href="group_9.html#E1:180"><i><font color="Green" title="E153">A1</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>thus </b><a NAME="E24:180_1_3"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E9:180_1_3"><i><font color="Green" title="E161">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><a NAME="E25:180_1_3"/>
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E15:180_1_3"><i><font color="Green" title="E163">A20</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><a NAME="E26:180_1_3"/>
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="group_9.html#T76" target="_self" title="GROUP_9:th.76">Th76</a>, <a class="txt" href="group_9.html#E23:180_1_3"><i><font color="Green" title="E165">A22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
