<?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>, <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>;<br/>







<b>assume </b><a NAME="E1:171"/><i><font color="Green" title="E144">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>
 ;<br/>

<b>assume </b><a NAME="E2:171"/><i><font color="Green" title="E145">A2</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a>
 ;<br/>

<b>assume </b><a NAME="E3:171"/><i><font color="Green" title="E146">A3</font></i>: 
( <font color="Maroon" title="c4">s2</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a> &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">s2</font> )
 ;<br/>

<a NAME="E4:171"/>
( <font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a> &amp; ( for <font color="Olive" title="b1">s2</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>  holds <br/>(  not <font color="Maroon" title="c3">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">s2</font> or  not <font color="Olive" title="b1">s2</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a> or  not <font color="Olive" title="b1">s2</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> ) ) )
 
<b>by </b><i><a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a>, <a class="txt" href="group_9.html#E2:171"><i><font color="Green" title="E145">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:171"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E1:171"><i><font color="Green" title="E144">A1</font></i></a>, <a class="txt" href="group_9.html#E3:171"><i><font color="Green" title="E146">A3</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>;<br/>


</div>
