<?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#R5" title="GROUP_9:pred.5">is_equivalent_with</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>  holds <br/><font color="Olive" title="b3">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> </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#R5" title="GROUP_9:pred.5">is_equivalent_with</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>  holds <br/><font color="Olive" title="b2">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> </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#R5" title="GROUP_9:pred.5">is_equivalent_with</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>  implies <font color="Maroon" title="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:193"/><span class="lab"><font color="Green" title="E166">A1</font></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>
 ; <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 <font color="Maroon" title="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:193"/><span class="lab"><font color="Green" title="E167">A2</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"> <font color="Maroon" title="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_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="E1:193_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1_1"><span class="kw">suppose </span></a><a NAME="E1:193_1_1"/><span class="lab"><font color="Green" title="E168">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a 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> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> </span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1_1_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="E1:193_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_1_1"><span class="lab"><font color="Green" title="E168">A3</font></span></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:193_1_1_1_1"/><span class="lab"><font color="Green" title="E169">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> </span><br/><div class="add"><a NAME="E2:193_1_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193"><span class="lab"><font color="Green" title="E166">A1</font></span></a>, <a class="ref" href="group_9.html#D35" target="_self" title="GROUP_9:def.35">Def35</a></span>;<br/><a NAME="E3:193_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A5</font></span>: 
<font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E4:193_1_1_1_1"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_1_1_1_1"><span class="lab"><font color="Green" title="E169">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:193_1_1_1_1"/>
<font color="Maroon" title="c4">s2</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="#E2:193"><span class="lab"><font color="Green" title="E167">A2</font></span></a>, <a class="txt" href="#E3:193_1_1_1_1"><span class="lab"><font color="Green" title="E170">A5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:193_1_1_1_2"/><span class="lab"><font color="Green" title="E169">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> </span><br/><div class="add"><a NAME="E2:193_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A7</font></span>: 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></span>;<br/><a NAME="E3:193_1_1_1_2"/><span class="lab"><font color="Green" title="E171">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193"><span class="lab"><font color="Green" title="E166">A1</font></span></a>, <a class="txt" href="#E1:193_1_1_1_2"><span class="lab"><font color="Green" title="E169">A6</font></span></a>, <a class="ref" href="group_9.html#D35" target="_self" title="GROUP_9:def.35">Def35</a></span>;<br/><font color="Maroon" title="c3">s1</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_1_1_1_2"><span class="lab"><font color="Green" title="E169">A6</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:193_1_1_1_2"><span class="lab"><font color="Green" title="E170">A7</font></span></a>, <a class="txt" href="#E3:193_1_1_1_2"><span class="lab"><font color="Green" title="E171">A8</font></span></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:193_1_1_1_2"><span class="lab"><font color="Green" title="E171">A8</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:193_1_1_1_2"/>
<font color="Maroon" title="c4">s2</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="#E2:193"><span class="lab"><font color="Green" title="E167">A2</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1_2"><span class="kw">suppose </span></a><a NAME="E1:193_1_2"/><span class="lab"><font color="Green" title="E168">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">s2</font> is  <a href="group_9.html#V10" title="GROUP_9:attr.10">jordan_holder</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">f2</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>;<br/><span class="kw">set </span><font color="Maroon" title="c6">f1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font>;<br/><a NAME="E2:193_1_2"/><span class="lab"><font color="Green" title="E169">A10</font></span>: 
not <font color="Maroon" title="c3">s1</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="#E1:193_1_2"><span class="lab"><font color="Green" title="E168">A9</font></span></a></span>;<br/><a NAME="E3:193_1_2"/><span class="lab"><font color="Green" title="E170">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193"><span class="lab"><font color="Green" title="E166">A1</font></span></a>, <a class="txt" href="#E1:193_1_2"><span class="lab"><font color="Green" title="E168">A9</font></span></a>, <a class="ref" href="group_9.html#D35" target="_self" title="GROUP_9:def.35">Def35</a></span>;<br/><a NAME="E4:193_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c4">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </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="c3">s1</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:193_1_2"/><span class="lab"><font color="Green" title="E171">A12</font></span>: 
 <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="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193"><span class="lab"><font color="Green" title="E166">A1</font></span></a>, <a class="txt" href="#E2:193_1_2"><span class="lab"><font color="Green" title="E169">A10</font></span></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</a></span>;<br/><a NAME="E7:193_1_2"/><span class="lab"><font color="Green" title="E172">A13</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="c3">s1</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="c4">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:193_1_2"><span class="lab"><font color="Green" title="E171">A12</font></span></a>, <a class="ref" href="group_9.html#D37" target="_self" title="GROUP_9:def.37">Def37</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="193_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">j</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" title="c8">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="c4">s2</font></span>)</span> implies <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="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">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="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">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="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>;<br/><span class="kw">assume </span><a NAME="E1:193_1_2_1"/><span class="lab"><font color="Green" title="E173">A14</font></span>: 
<font color="Maroon" title="c8">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="c4">s2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font></span><br/><a NAME="E2:193_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A15</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="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c4">s2</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="lab"><font color="Green" title="E175">A16</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
<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 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="c4">s2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:193_1_2"><span class="lab"><font color="Green" title="E171">A12</font></span></a>, <a class="ref" href="group_9.html#D37" target="_self" title="GROUP_9:def.37">Def37</a></span>
<br/>.= 
 <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="c4">s2</font></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="E4:193_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A17</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c4">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_1_2_1"><span class="lab"><font color="Green" title="E173">A14</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E6:193_1_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</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="c4">s2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:193_1_2_1"><span class="lab"><font color="Green" title="E176">A17</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="E7:193_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A18</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:193_1_2"><span class="lab"><font color="Green" title="E172">A13</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="E8:193_1_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="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <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="c13">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="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span>, <font color="Maroon" title="c14">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="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="#E2:193_1_2_1"><span class="lab"><font color="Green" title="E174">A15</font></span></a>, <a class="ref" href="group_9.html#T102" target="_self" title="GROUP_9:th.102">Th102</a></span>;<br/><a NAME="E10:193_1_2_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c12">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_1_2_1"><span class="lab"><font color="Green" title="E173">A14</font></span></a>, <a class="txt" href="#E3:193_1_2_1"><span class="lab"><font color="Green" title="E175">A16</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/><a NAME="E11:193_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A19</font></span>: 
<font color="Maroon" title="c13">H1</font>,<font color="Maroon" title="c14">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="#E6:193_1_2"><span class="lab"><font color="Green" title="E171">A12</font></span></a>, <a class="ref" href="group_9.html#D37" target="_self" title="GROUP_9:def.37">Def37</a></span>;<br/><a NAME="E12:193_1_2_1"/>
<font color="Maroon" title="c13">H1</font> is   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</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="#E2:193"><span class="lab"><font color="Green" title="E167">A2</font></span></a>, <a class="txt" href="#E1:193_1_2"><span class="lab"><font color="Green" title="E168">A9</font></span></a>, <a class="txt" href="#E7:193_1_2_1"><span class="lab"><font color="Green" title="E177">A18</font></span></a>, <a class="ref" href="group_9.html#T110" target="_self" title="GROUP_9:th.110">Th110</a></span>;<br/><span class="kw">hence </span><a NAME="E13:193_1_2_1"/>
<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="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is   <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>   <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</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="#E11:193_1_2_1"><span class="lab"><font color="Green" title="E178">A19</font></span></a>, <a class="ref" href="group_9.html#T82" target="_self" title="GROUP_9:th.82">Th82</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="E9:193_1_2"/>
<font color="Maroon" title="c4">s2</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="#E3:193_1_2"><span class="lab"><font color="Green" title="E170">A11</font></span></a>, <a class="ref" href="group_9.html#T110" target="_self" title="GROUP_9:th.110">Th110</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>
