<?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> 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="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</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> 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="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</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> 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="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:201"/><span class="lab"><font color="Green" title="E173">A1</font></span>: 
<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="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:201"/><span class="lab"><font color="Green" title="E174">A2</font></span>: 
<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:()"><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#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="201_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:201_1"/>
( <font color="Maroon" title="c3">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c3">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="201_1_1"><span class="kw">suppose </span></a><a NAME="E1:201_1_1"/><span class="lab"><font color="Green" title="E175">A3</font></span>: 
<font color="Maroon" title="c3">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c6">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ; <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="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">s2</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">s2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c6">x</font></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E1:201_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>; <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="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">s2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/><span class="kw">thus </span><a NAME="E2:201_1_1_1_1"/>
<font color="Maroon" title="c3">s1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">s2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_1"><span class="lab"><font color="Green" title="E175">A3</font></span></a>, <a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</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><span class="kw">end;</span></div><a NAME="E2:201_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A4</font></span>: 
<font color="Maroon" title="c4">s2</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def32</a></span>;<br/><span class="kw">assume </span><a NAME="E3:201_1_1_1"/><span class="lab"><font color="Green" title="E177">A5</font></span>: 
not <font color="Maroon" title="c4">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:201_1_1_1"/>
<font color="Maroon" title="c4">s2</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="txt" href="#E2:201"><span class="lab"><font color="Green" title="E174">A2</font></span></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></span>;<br/><span class="kw">hence </span><a NAME="E5:201_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201"><span class="lab"><font color="Green" title="E173">A1</font></span></a>, <a class="txt" href="#E1:201_1_1"><span class="lab"><font color="Green" title="E175">A3</font></span></a>, <a class="txt" href="#E3:201_1_1_1"><span class="lab"><font color="Green" title="E177">A5</font></span></a>, <a class="txt" href="#E2:201_1_1_1"><span class="lab"><font color="Green" title="E176">A4</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:201_1_1"/>
<font color="Maroon" title="c3">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_1"><span class="lab"><font color="Green" title="E175">A3</font></span></a>, <a class="ref" href="group_9.html#T107" target="_self" title="GROUP_9:th.107">Th107</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="201_1_2"><span class="kw">suppose </span></a><a NAME="E1:201_1_2"/><span class="lab"><font color="Green" title="E175">A6</font></span>: 
not <font color="Maroon" title="c3">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">s19</font>, <font color="Olive" title="b2">s29</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>  st not <font color="Olive" title="b1">s19</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Olive" title="b2">s29</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s19</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> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 &amp; <font color="Olive" title="b1">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b2">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b3">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="Olive" title="b1">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b1">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b2">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c1">O</font> holds <br/> ex <font color="Olive" title="b3">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c1">O</font>;<br/><div><span class="lab"><font color="Green" title="E176">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:201_1_2_1"/><span class="lab"><font color="Green" title="E177">A8</font></span>: 
<font color="Maroon" title="c4">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c6">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ; <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="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">s2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">s1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c6">x</font></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E1:201_1_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>; <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="c4">s2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">s1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/><span class="kw">thus </span><a NAME="E2:201_1_2_1_1"/>
<font color="Maroon" title="c4">s2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">s1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_2_1"><span class="lab"><font color="Green" title="E177">A8</font></span></a>, <a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</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><span class="kw">end;</span></div><a NAME="E3:201_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def32</a></span>;<br/><a NAME="E4:201_1_2_1"/>
<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="txt" href="#E1:201"><span class="lab"><font color="Green" title="E173">A1</font></span></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></span>;<br/><span class="kw">hence </span><a NAME="E5:201_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:201"><span class="lab"><font color="Green" title="E174">A2</font></span></a>, <a class="txt" href="#E1:201_1_2"><span class="lab"><font color="Green" title="E175">A6</font></span></a>, <a class="txt" href="#E1:201_1_2_1"><span class="lab"><font color="Green" title="E177">A8</font></span></a>, <a class="txt" href="#E3:201_1_2_1"><span class="lab"><font color="Green" title="E178">A9</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><span class="kw">end;</span></div><a NAME="E3:201_1_2"/><span class="lab"><font color="Green" title="E177">A10</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:201_1_2_2"/><span class="lab"><font color="Green" title="E178">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">s19</font>, <font color="Maroon" title="c7">s29</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"> ( not <font color="Maroon" title="c6">s19</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c7">s29</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:201_1_2_2_1"/>
not <font color="Maroon" title="c6">s19</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E2:201_1_2_2_1"/>
not <font color="Maroon" title="c7">s29</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E3:201_1_2_2_1"/><span class="lab"><font color="Green" title="E179">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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="c6">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">set </span><font color="Maroon" title="c8">f1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s19</font>;<br/><span class="kw">assume </span><a NAME="E4:201_1_2_2_1"/><span class="lab"><font color="Green" title="E180">A13</font></span>: 
<font color="Maroon" title="c6">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</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="c7">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><a NAME="E5:201_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:201_1_2_2_1"/><span class="lab"><font color="Green" title="E181">A14</font></span>: 
<font color="Maroon" title="c9">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="c6">s19</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:201_1_2_2_1"/><span class="lab"><font color="Green" title="E182">A15</font></span>: 
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="c6">s19</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201"><span class="lab"><font color="Green" title="E173">A1</font></span></a>, <a class="txt" href="#E3:201_1_2_2_1"><span class="lab"><font color="Green" title="E179">A12</font></span></a>, <a class="txt" href="#E4:201_1_2_2_1"><span class="lab"><font color="Green" title="E180">A13</font></span></a>, <a class="ref" href="group_9.html#T109" target="_self" title="GROUP_9:th.109">Th109</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">s199</font> =  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c6">s19</font>,<font color="Maroon" title="c9">i</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T114" title="FINSEQ_3:th.114">FINSEQ_3:114</a></span>;<br/><a NAME="E10:201_1_2_2_1"/><span class="lab"><font color="Green" title="E183">A16</font></span>: 
<font color="Maroon" title="c9">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="c6">s19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:201_1_2_2_1"><span class="lab"><font color="Green" title="E181">A14</font></span></a>, <a class="txt" href="#E8:201_1_2_2_1"><span class="lab"><font color="Green" title="E182">A15</font></span></a>, <a class="ref" href="group_9.html#T103" target="_self" title="GROUP_9:th.103">Th103</a></span>;<br/><a NAME="E11:201_1_2_2_1"/><span class="lab"><font color="Green" title="E184">A17</font></span>: 
( <font color="Maroon" title="c9">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="c6">s19</font> &amp; <font color="Maroon" title="c6">s19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:201_1_2_2_1"><span class="lab"><font color="Green" title="E181">A14</font></span></a>, <a class="txt" href="#E8:201_1_2_2_1"><span class="lab"><font color="Green" title="E182">A15</font></span></a>, <a class="ref" href="group_9.html#T103" target="_self" title="GROUP_9:th.103">Th103</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">s199</font> = <font color="Maroon" title="c10">s199</font> as   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:201_1_2_2_1"><span class="lab"><font color="Green" title="E183">A16</font></span></a>, <a class="ref" href="group_9.html#T94" target="_self" title="GROUP_9:th.94">Th94</a></span>;<br/><a NAME="E13:201_1_2_2_1"/><span class="lab"><font color="Green" title="E185">A18</font></span>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c11">s199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</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="c6">s19</font></span>)</span>,<font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:201_1_2_2_1"><span class="lab"><font color="Green" title="E183">A16</font></span></a>, <a class="txt" href="#E11:201_1_2_2_1"><span class="lab"><font color="Green" title="E184">A17</font></span></a>, <a class="ref" href="group_9.html#T104" target="_self" title="GROUP_9:th.104">Th104</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">f2</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s29</font>;<br/><span class="kw">assume </span><a NAME="E14:201_1_2_2_1"/><span class="lab"><font color="Green" title="E186">A19</font></span>: 
<font color="Maroon" title="c7">s29</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"> (  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">given </span><font color="Maroon" title="c13">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">s19</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E16:201_1_2_2_1"/><span class="lab"><font color="Green" title="E187">A20</font></span>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Maroon" title="c13">p</font>,<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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">set </span><font color="Maroon" title="c14">H1</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="c6">s19</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>;<br/><a NAME="E17:201_1_2_2_1"/><span class="lab"><font color="Green" title="E188">A21</font></span>: 
<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="c6">s19</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c6">s19</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:201_1_2_2_1"><span class="lab"><font color="Green" title="E181">A14</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">j</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">j</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c17">H2</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="c7">s29</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">s299</font> =  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c7">s29</font>,<font color="Maroon" title="c16">j</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T114" title="FINSEQ_3:th.114">FINSEQ_3:114</a></span>;<br/><a NAME="E20:201_1_2_2_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c6">s19</font></span>)</span>
 ;<br/><a NAME="E21:201_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A22</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c6">s19</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E22:201_1_2_2_1"/><span class="lab"><font color="Green" title="E190">A23</font></span>: 
 <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="c6">s19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">s29</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:201_1_2_2_1"><span class="lab"><font color="Green" title="E187">A20</font></span></a>, <a class="ref" href="group_9.html#D37" target="_self" title="GROUP_9:def.37">Def37</a></span>;<br/><a NAME="E23:201_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:201_1_2_2_1"><span class="lab"><font color="Green" title="E181">A14</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/><a NAME="E24:201_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c6">s19</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:201_1_2_2_1"><span class="lab"><font color="Green" title="E189">A22</font></span></a></span>;<br/><a NAME="E25:201_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E191">A24</font></span>: 
<font color="Maroon" title="c16">j</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="c7">s29</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:201_1_2_2_1"><span class="lab"><font color="Green" title="E190">A23</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E26:201_1_2_2_1"/><span class="kw">then </span>
<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="c7">s29</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</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="c7">s29</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">H1</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="c6">s19</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c20">H2</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="c7">s29</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:201_1_2_2_1"><span class="lab"><font color="Green" title="E188">A21</font></span></a>, <a class="ref" href="group_9.html#T102" target="_self" title="GROUP_9:th.102">Th102</a></span>;<br/><a NAME="E28:201_1_2_2_1"/><span class="lab"><font color="Green" title="E192">A25</font></span>: 
<font color="Maroon" title="c19">H1</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:201_1_2_2_1"><span class="lab"><font color="Green" title="E182">A15</font></span></a></span>;<br/><a NAME="E29:201_1_2_2_1"/>
<font color="Maroon" title="c19">H1</font>,<font color="Maroon" title="c20">H2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:201_1_2_2_1"><span class="lab"><font color="Green" title="E187">A20</font></span></a>, <a class="txt" href="#E7:201_1_2_2_1"><span class="lab"><font color="Green" title="E181">A14</font></span></a>, <a class="ref" href="group_9.html#D37" target="_self" title="GROUP_9:def.37">Def37</a></span>;<br/><a NAME="E30:201_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E193">A26</font></span>: 
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="c7">s29</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E28:201_1_2_2_1"><span class="lab"><font color="Green" title="E192">A25</font></span></a>, <a class="ref" href="group_9.html#T58" target="_self" title="GROUP_9:th.58">Th58</a></span>;<br/><a NAME="E31:201_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E194">A27</font></span>: 
( <font color="Maroon" title="c16">j</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="c7">s29</font> &amp; <font color="Maroon" title="c16">j</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="c7">s29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:201_1_2_2_1"><span class="lab"><font color="Green" title="E191">A24</font></span></a>, <a class="ref" href="group_9.html#T103" target="_self" title="GROUP_9:th.103">Th103</a></span>;<br/><a NAME="E32:201_1_2_2_1"/><span class="lab"><font color="Green" title="E195">A28</font></span>: 
<font color="Maroon" title="c7">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:201_1_2_2_1"><span class="lab"><font color="Green" title="E191">A24</font></span></a>, <a class="txt" href="#E30:201_1_2_2_1"><span class="lab"><font color="Green" title="E193">A26</font></span></a>, <a class="ref" href="group_9.html#T103" target="_self" title="GROUP_9:th.103">Th103</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">s299</font> = <font color="Maroon" title="c18">s299</font> as   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:201_1_2_2_1"><span class="lab"><font color="Green" title="E194">A27</font></span></a>, <a class="ref" href="group_9.html#T94" target="_self" title="GROUP_9:th.94">Th94</a></span>;<br/><a NAME="E34:201_1_2_2_1"/><span class="lab"><font color="Green" title="E196">A29</font></span>: 
( <font color="Maroon" title="c21">s299</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp; not <font color="Maroon" title="c21">s299</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:201"><span class="lab"><font color="Green" title="E174">A2</font></span></a>, <a class="txt" href="#E2:201_1_2"><span class="lab"><font color="Green" title="E176">A7</font></span></a>, <a class="txt" href="#E14:201_1_2_2_1"><span class="lab"><font color="Green" title="E186">A19</font></span></a>, <a class="txt" href="#E31:201_1_2_2_1"><span class="lab"><font color="Green" title="E194">A27</font></span></a>, <a class="txt" href="#E32:201_1_2_2_1"><span class="lab"><font color="Green" title="E195">A28</font></span></a>, <a class="ref" href="group_9.html#T97" target="_self" title="GROUP_9:th.97">Th97</a>, <a class="ref" href="group_9.html#T99" target="_self" title="GROUP_9:th.99">Th99</a></span>;<br/><a NAME="E35:201_1_2_2_1"/><span class="lab"><font color="Green" title="E197">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s199</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> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:201_1_2_2_1"><span class="lab"><font color="Green" title="E179">A12</font></span></a>, <a class="txt" href="#E10:201_1_2_2_1"><span class="lab"><font color="Green" title="E183">A16</font></span></a>, <a class="ref" href="finseq_3.html#T118" title="FINSEQ_3:th.118">FINSEQ_3:118</a></span>;<br/><a NAME="E36:201_1_2_2_1"/>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c21">s299</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</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="c7">s29</font></span>)</span>,<font color="Maroon" title="c16">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:201_1_2_2_1"><span class="lab"><font color="Green" title="E194">A27</font></span></a>, <a class="txt" href="#E32:201_1_2_2_1"><span class="lab"><font color="Green" title="E195">A28</font></span></a>, <a class="ref" href="group_9.html#T104" target="_self" title="GROUP_9:th.104">Th104</a></span>;<br/><a NAME="E37:201_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E198">A31</font></span>: 
 ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c11">s199</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c11">s199</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c21">s299</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:201_1_2_2_1"><span class="lab"><font color="Green" title="E187">A20</font></span></a>, <a class="txt" href="#E7:201_1_2_2_1"><span class="lab"><font color="Green" title="E181">A14</font></span></a>, <a class="txt" href="#E13:201_1_2_2_1"><span class="lab"><font color="Green" title="E185">A18</font></span></a>, <a class="ref" href="group_9.html#T106" target="_self" title="GROUP_9:th.106">Th106</a></span>;<br/><a NAME="E38:201_1_2_2_1"/>
( <font color="Maroon" title="c11">s199</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> &amp; not <font color="Maroon" title="c11">s199</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201"><span class="lab"><font color="Green" title="E173">A1</font></span></a>, <a class="txt" href="#E1:201_1_2"><span class="lab"><font color="Green" title="E175">A6</font></span></a>, <a class="txt" href="#E4:201_1_2_2_1"><span class="lab"><font color="Green" title="E180">A13</font></span></a>, <a class="txt" href="#E10:201_1_2_2_1"><span class="lab"><font color="Green" title="E183">A16</font></span></a>, <a class="txt" href="#E11:201_1_2_2_1"><span class="lab"><font color="Green" title="E184">A17</font></span></a>, <a class="ref" href="group_9.html#T97" target="_self" title="GROUP_9:th.97">Th97</a>, <a class="ref" href="group_9.html#T99" target="_self" title="GROUP_9:th.99">Th99</a></span>;<br/><span class="kw">hence </span><a NAME="E39:201_1_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_2_2"><span class="lab"><font color="Green" title="E178">A11</font></span></a>, <a class="txt" href="#E35:201_1_2_2_1"><span class="lab"><font color="Green" title="E197">A30</font></span></a>, <a class="txt" href="#E34:201_1_2_2_1"><span class="lab"><font color="Green" title="E196">A29</font></span></a>, <a class="txt" href="#E37:201_1_2_2_1"><span class="lab"><font color="Green" title="E198">A31</font></span></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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:201_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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><span class="kw">end;</span></div><a NAME="E4:201_1_2"/><span class="lab"><font color="Green" title="E178">A32</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="201_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">s19</font>, <font color="Maroon" title="c6">s29</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"> ( not <font color="Maroon" title="c5">s19</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c6">s29</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s19</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> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c5">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> implies  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:201_1_2_3"/><span class="lab"><font color="Green" title="E179">A33</font></span>: 
( not <font color="Maroon" title="c5">s19</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c6">s29</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s19</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> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c5">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> or  not <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> or for <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c5">s19</font></span>)</span></span>)</span>  holds not  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> or  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:201_1_2_3"/><span class="lab"><font color="Green" title="E180">A34</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s19</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> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</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="c6">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> or for <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c5">s19</font></span>)</span></span>)</span>  holds not  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> or  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:201_1_2_3"/><span class="lab"><font color="Green" title="E181">A35</font></span>: 
<font color="Maroon" title="c6">s29</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"> ( for <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c5">s19</font></span>)</span></span>)</span>  holds not  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> or  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c5">s19</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E5:201_1_2_3"/><span class="lab"><font color="Green" title="E182">A36</font></span>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Maroon" title="c7">p</font>,<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">  ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font></span><br/>

<a NAME="E6:201_1_2_3"/><span class="lab"><font color="Green" title="E183">A37</font></span>: 
<font color="Maroon" title="c5">s19</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c6">s29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_2_3"><span class="lab"><font color="Green" title="E179">A33</font></span></a>, <a class="txt" href="#E5:201_1_2_3"><span class="lab"><font color="Green" title="E182">A36</font></span></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</a></span>;<br/>
<a NAME="E7:201_1_2_3"/>
<font color="Maroon" title="c5">s19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:201_1_2_3"><span class="lab"><font color="Green" title="E180">A34</font></span></a>, <a class="ref" href="group_9.html#T96" target="_self" title="GROUP_9:th.96">Th96</a></span>;<br/>
<a NAME="E8:201_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">s29</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201"><span class="lab"><font color="Green" title="E173">A1</font></span></a>, <a class="txt" href="#E6:201_1_2_3"><span class="lab"><font color="Green" title="E183">A37</font></span></a>, <a class="ref" href="group_9.html#T115" target="_self" title="GROUP_9:th.115">Th115</a></span>;<br/>
<a NAME="E9:201_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">s29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:201"><span class="lab"><font color="Green" title="E174">A2</font></span></a>, <a class="txt" href="#E3:201_1_2_3"><span class="lab"><font color="Green" title="E181">A35</font></span></a>, <a class="ref" href="group_9.html#T98" target="_self" title="GROUP_9:th.98">Th98</a></span>;<br/>
<a NAME="E10:201_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:201_1_2_3"><span class="lab"><font color="Green" title="E180">A34</font></span></a>, <a class="txt" href="#E6:201_1_2_3"><span class="lab"><font color="Green" title="E183">A37</font></span></a>, <a class="ref" href="group_9.html#T96" target="_self" title="GROUP_9:th.96">Th96</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:201_1_2_3"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_2"><span class="lab"><font color="Green" title="E175">A6</font></span></a>, <a class="txt" href="#E2:201_1_2"><span class="lab"><font color="Green" title="E176">A7</font></span></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</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><span class="kw">end;</span></div><a NAME="E5:201_1_2"/><span class="lab"><font color="Green" title="E179">A38</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:201_1_2"><span class="lab"><font color="Green" title="E178">A32</font></span></a>, <a class="txt" href="#E3:201_1_2"><span class="lab"><font color="Green" title="E177">A10</font></span></a>);<br/></span><span class="kw">consider </span><font color="Maroon" title="c5">s19</font>, <font color="Maroon" title="c6">s29</font> being   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E7:201_1_2"/><span class="lab"><font color="Green" title="E180">A39</font></span>: 
<font color="Maroon" title="c5">s19</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font>
 <span class="kw">and </span><br/><a NAME="E8:201_1_2"/><span class="lab"><font color="Green" title="E181">A40</font></span>: 
<font color="Maroon" title="c6">s29</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">and </span><br/><a NAME="E9:201_1_2"/><span class="lab"><font color="Green" title="E182">A41</font></span>: 
<font color="Maroon" title="c5">s19</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c6">s29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#T118" target="_self" title="GROUP_9:th.118">Th118</a></span>;<br/><a NAME="E10:201_1_2"/><span class="lab"><font color="Green" title="E183">A42</font></span>: 
not <font color="Maroon" title="c5">s19</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_2"><span class="lab"><font color="Green" title="E175">A6</font></span></a>, <a class="txt" href="#E7:201_1_2"><span class="lab"><font color="Green" title="E180">A39</font></span></a>, <a class="ref" href="group_9.html#T97" target="_self" title="GROUP_9:th.97">Th97</a></span>;<br/><a NAME="E11:201_1_2"/><span class="lab"><font color="Green" title="E184">A43</font></span>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s19</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> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:201_1_2"><span class="lab"><font color="Green" title="E180">A39</font></span></a>, <a class="ref" href="group_9.html#T95" target="_self" title="GROUP_9:th.95">Th95</a></span>;<br/><a NAME="E12:201_1_2"/><span class="lab"><font color="Green" title="E185">A44</font></span>: 
not <font color="Maroon" title="c6">s29</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:201_1_2"><span class="lab"><font color="Green" title="E176">A7</font></span></a>, <a class="txt" href="#E8:201_1_2"><span class="lab"><font color="Green" title="E181">A40</font></span></a>, <a class="ref" href="group_9.html#T97" target="_self" title="GROUP_9:th.97">Th97</a></span>;<br/><a NAME="E13:201_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c5">s19</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s19</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">s29</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p9</font>,<font color="Maroon" title="c1">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:201_1_2"><span class="lab"><font color="Green" title="E182">A41</font></span></a>, <a class="txt" href="#E10:201_1_2"><span class="lab"><font color="Green" title="E183">A42</font></span></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</a></span>;<br/><a NAME="E14:201_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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> st  <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:201_1_2"><span class="lab"><font color="Green" title="E180">A39</font></span></a>, <a class="txt" href="#E8:201_1_2"><span class="lab"><font color="Green" title="E181">A40</font></span></a>, <a class="txt" href="#E10:201_1_2"><span class="lab"><font color="Green" title="E183">A42</font></span></a>, <a class="txt" href="#E12:201_1_2"><span class="lab"><font color="Green" title="E185">A44</font></span></a>, <a class="txt" href="#E5:201_1_2"><span class="lab"><font color="Green" title="E179">A38</font></span></a>, <a class="txt" href="#E11:201_1_2"><span class="lab"><font color="Green" title="E184">A43</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:201_1_2"/>
<font color="Maroon" title="c3">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:201_1_2"><span class="lab"><font color="Green" title="E175">A6</font></span></a>, <a class="txt" href="#E2:201_1_2"><span class="lab"><font color="Green" title="E176">A7</font></span></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
