<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b2">s1</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Olive" title="b3">s2</font> &amp; <font color="Olive" title="b3">s2</font> is <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">s2</font> holds <br/> ex <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b4">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="Olive" title="b2">s1</font></span>)</span> &amp; ( for <font color="Olive" title="b5">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="b5">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="Olive" title="b2">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b5">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">s1</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b2">s2</font> is <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s2</font> holds <br/> ex <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b3">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="Olive" title="b1">s1</font></span>)</span> &amp; ( for <font color="Olive" title="b4">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="b4">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="Olive" title="b1">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b4">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">s1</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp; <font color="Maroon" title="c4">s2</font> is <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> &amp;  <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> implies  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) ) )</span><br/>







<b>assume </b><a NAME="E1:182"/><i><font color="Green" title="E155">A1</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">s2</font> is <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> or  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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> or  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) ) )</span><br/>

<b>assume </b><a NAME="E2:182"/><i><font color="Green" title="E156">A2</font></i>: 
<font color="Maroon" title="c4">s2</font> is <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> or  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) ) )</span><br/>

<b>assume </b><a NAME="E3:182"/><i><font color="Green" title="E157">A3</font></i>: 
 <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )</span><br/>

<a NAME="E4:182"/><b>then </b>
not <font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V9" title="GROUP_9:attr.9">strictly_decreasing</a>
 
<b>by </b><i><a class="txt" href="#E1:182"><i><font color="Green" title="E155">A1</font></i></a>, <a class="txt" href="#E2:182"><i><font color="Green" title="E156">A2</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>;<br/>
<a NAME="E5:182"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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;  ex <font color="Olive" title="b2">H1</font> being   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">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="b2">H1</font> st <br/>( <font color="Olive" title="b2">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="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )
 
<b>by </b><i><a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c6">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c7">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="Maroon" title="c6">H1</font><b> such that </b><br/><a NAME="E7:182"/><i><font color="Green" title="E158">A4</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>and </b><br/><a NAME="E8:182"/><i><font color="Green" title="E159">A5</font></i>: 
( <font color="Maroon" title="c6">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="c5">i</font> &amp; <font color="Maroon" title="c7">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="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E9:182"/><i><font color="Green" title="E160">A6</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="c7">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>
<a NAME="E10:182"/>
<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="txt" href="#E7:182"><i><font color="Green" title="E158">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="E11:182"/><b>then </b><i><font color="Green" title="E161">A7</font></i>: 
<font color="Maroon" title="c5">i</font> <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>  <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="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:182"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <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="E13:182"/><b>then </b><i><font color="Green" title="E162">A8</font></i>: 
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="txt" href="#E11:182"><i><font color="Green" title="E161">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:182_1"/>
(  <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#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or  <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 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_1_1"><b>suppose </b></a><a NAME="E1:182_1_1"/>
 <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#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )</span><br/><div class="add"><a NAME="E2:182_1_1"/><b>then </b><i><font color="Green" title="E163">A9</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> 1
 <b>by </b><i><a class="txt" href="#E13:182"><i><font color="Green" title="E162">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> implies 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="c8">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> not <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> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/><b>assume </b><a NAME="E1:182_1_1_1"/>
<font color="Maroon" title="c8">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> implies 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="c8">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> not <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> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/><a NAME="E2:182_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">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="#E2:182_1_1"><i><font color="Green" title="E163">A9</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="E3:182_1_1_1"/><b>then </b><i><font color="Green" title="E164">A10</font></i>: 
<font color="Maroon" title="c8">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/><b>assume </b><a NAME="E4:182_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c8">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> not <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> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/><a NAME="E5:182_1_1_1"/><b>then </b><i><font color="Green" title="E165">A11</font></i>: 
2 <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="#E2:182_1_1"><i><font color="Green" title="E163">A9</font></i></a>, <a class="txt" href="#E3:182_1_1_1"><i><font color="Green" title="E164">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>let </b><font color="Maroon" title="c9">H1</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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="Maroon" title="c9">H1</font>  st <font color="Maroon" title="c9">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="c8">i</font> &amp; <font color="Olive" title="b1">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> not <font color="Maroon" title="c9">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/><b>let </b><font color="Maroon" title="c10">N1</font> be  <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="c9">H1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">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="c8">i</font> &amp; <font color="Maroon" title="c10">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies  not <font color="Maroon" title="c9">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c10">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/><b>assume </b><a NAME="E6:182_1_1_1"/>
<font color="Maroon" title="c9">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="c8">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies  not <font color="Maroon" title="c9">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c10">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/><b>assume </b><a NAME="E7:182_1_1_1"/>
<font color="Maroon" title="c10">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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c9">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c10">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/><b>assume </b><a NAME="E8:182_1_1_1"/>
<font color="Maroon" title="c9">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c10">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>thus </b><a NAME="E9:182_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:182_1_1_1"><i><font color="Green" title="E165">A11</font></i></a>, <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E4:182_1_1"/><b>then </b>
<font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V9" title="GROUP_9:attr.9">strictly_decreasing</a>
 <b>by </b><i><a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a></i>;<br/><b>hence </b><a NAME="E5:182_1_1"/>
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )
 <b>by </b><i><a class="txt" href="#E1:182"><i><font color="Green" title="E155">A1</font></i></a>, <a class="txt" href="#E2:182"><i><font color="Green" title="E156">A2</font></i></a>, <a class="txt" href="#E3:182"><i><font color="Green" title="E157">A3</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_1_2"><b>suppose </b></a><a NAME="E1:182_1_2"/><i><font color="Green" title="E163">A12</font></i>: 
 <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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )</span><br/><div class="add"><a NAME="E2:182_1_2"/><b>then </b><i><font color="Green" title="E164">A13</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#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> holds <br/>for <font color="Olive" title="b2">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="b3">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="b2">H1</font>  st <font color="Olive" title="b2">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="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N1</font> ) )
 <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="E3:182_1_2"/>
<font color="Maroon" title="c5">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="txt" href="#E7:182"><i><font color="Green" title="E158">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="E4:182_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <b>by </b><i><a class="txt" href="#E11:182"><i><font color="Green" title="E161">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:182_1_2"/><b>then </b><i><font color="Green" title="E165">A14</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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="txt" href="#E2:182_1_2"><i><font color="Green" title="E164">A13</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> &amp; ( 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="c5">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> ) )</span><br/><b>thus </b><a NAME="E6:182_1_2"/><i><font color="Green" title="E166">A15</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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="#E5:182_1_2"><i><font color="Green" title="E165">A14</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c5">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></span><br/><b>let </b><font color="Maroon" title="c8">H</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">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="c5">i</font> implies <font color="Maroon" title="c8">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/><b>assume </b><a NAME="E7:182_1_2"/>
<font color="Maroon" title="c8">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="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/><b>hence </b><a NAME="E8:182_1_2"/>
<font color="Maroon" title="c8">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="#E8:182"><i><font color="Green" title="E159">A5</font></i></a>, <a class="txt" href="#E9:182"><i><font color="Green" title="E160">A6</font></i></a>, <a class="txt" href="#E1:182_1_2"><i><font color="Green" title="E163">A12</font></i></a>, <a class="txt" href="#E6:182_1_2"><i><font color="Green" title="E166">A15</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
