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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="b2">s1</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  &amp; <font color="Olive" title="b3">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  holds <br/><font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s2</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="b1">s1</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  &amp; <font color="Olive" title="b2">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  holds <br/><font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c3">s1</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  &amp; <font color="Maroon" title="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  implies <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:167"/><span class="lab"><font color="Green" title="E141">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">s1</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  or  not <font color="Maroon" title="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  or <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:167"/>
<font color="Maroon" title="c3">s1</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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 <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<a NAME="E3:167"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A2</font></span>: 
<font color="Maroon" title="c3">s1</font> is  <a href="group_9.html#V9" title="GROUP_9:attr.9">strictly_decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:167"/>
( <font color="Maroon" title="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E5:167"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E141">A1</font></span></a>, <a class="txt" href="#E3:167"><span class="lab"><font color="Green" title="E142">A2</font></span></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
