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<div class="add">

<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>;<br/>







<b>assume </b><a NAME="E1:205"/><i><font color="Green" title="E177">A1</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a>
 ;<br/>

<b>assume </b><a NAME="E2:205"/><i><font color="Green" title="E178">A2</font></i>: 
<font color="Maroon" title="c4">s2</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="205_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:205_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="205_1_1"><b>suppose </b></a><a NAME="E1:205_1_1"/><i><font color="Green" title="E179">A3</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="205_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:205_1_1_1"/><i><font color="Green" title="E180">A4</font></i>: 
not <font color="Maroon" title="c4">s2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:205_1_1_1"/><i><font color="Green" title="E181">A5</font></i>: 
<font color="Maroon" title="c4">s2</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a>
 <b>by </b><i><a class="txt" href="group_9.html#E2:205"><i><font color="Green" title="E178">A2</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="205_1_1_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c5">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><b>take </b><font color="Maroon" title="c6">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><b>thus </b><a NAME="E1:205_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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>thus </b><a NAME="E2:205_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>
 <b>by </b><i><a class="txt" href="group_9.html#E1:205_1_1"><i><font color="Green" title="E179">A3</font></i></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</a></i>;<br/></div><b>end;</b></div><a NAME="E4:205_1_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def32</a></i>;<br/><b>hence </b><a NAME="E5:205_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a>, <a class="txt" href="group_9.html#E1:205"><i><font color="Green" title="E177">A1</font></i></a>, <a class="txt" href="group_9.html#E1:205_1_1"><i><font color="Green" title="E179">A3</font></i></a>, <a class="txt" href="group_9.html#E1:205_1_1_1"><i><font color="Green" title="E180">A4</font></i></a>, <a class="txt" href="group_9.html#E2:205_1_1_1"><i><font color="Green" title="E181">A5</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:205_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>
 <b>by </b><i><a class="ref" href="group_9.html#T107" target="_self" title="GROUP_9:th.107">Th107</a>, <a class="txt" href="group_9.html#E1:205_1_1"><i><font color="Green" title="E179">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="205_1_2"><b>suppose </b></a><a NAME="E1:205_1_2"/><i><font color="Green" title="E179">A6</font></i>: 
not <font color="Maroon" title="c3">s1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><div><i><font color="Green" title="E180">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="205_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:205_1_2_1"/><i><font color="Green" title="E181">A8</font></i>: 
<font color="Maroon" title="c4">s2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:205_1_2_1"/><i><font color="Green" title="E182">A9</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a>
 <b>by </b><i><a class="txt" href="group_9.html#E1:205"><i><font color="Green" title="E177">A1</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="205_1_2_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c5">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><b>take </b><font color="Maroon" title="c6">x</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><b>thus </b><a NAME="E1:205_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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>thus </b><a NAME="E2:205_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>
 <b>by </b><i><a class="txt" href="group_9.html#E1:205_1_2_1"><i><font color="Green" title="E181">A8</font></i></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</a></i>;<br/></div><b>end;</b></div><a NAME="E4:205_1_2_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def32</a></i>;<br/><b>hence </b><a NAME="E5:205_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a>, <a class="txt" href="group_9.html#E2:205"><i><font color="Green" title="E178">A2</font></i></a>, <a class="txt" href="group_9.html#E1:205_1_2"><i><font color="Green" title="E179">A6</font></i></a>, <a class="txt" href="group_9.html#E1:205_1_2_1"><i><font color="Green" title="E181">A8</font></i></a>, <a class="txt" href="group_9.html#E2:205_1_2_1"><i><font color="Green" title="E182">A9</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c5">s1'</font>, <font color="Maroon" title="c6">s2'</font> being   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:205_1_2"/><i><font color="Green" title="E181">A10</font></i>: 
( <font color="Maroon" title="c5">s1'</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">s2'</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font> &amp; <font color="Maroon" title="c5">s1'</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c6">s2'</font> )
 <b>by </b><i><a class="ref" href="group_9.html#T118" target="_self" title="GROUP_9:th.118">Th118</a></i>;<br/><a NAME="E5:205_1_2"/><i><font color="Green" title="E182">A11</font></i>: 
( not <font color="Maroon" title="c5">s1'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c6">s2'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="group_9.html#E4:205_1_2"><i><font color="Green" title="E181">A10</font></i></a>, <a class="txt" href="group_9.html#E1:205_1_2"><i><font color="Green" title="E179">A6</font></i></a>, <a class="txt" href="group_9.html#E2:205_1_2"><i><font color="Green" title="E180">A7</font></i></a>, <a class="ref" href="group_9.html#T97" target="_self" title="GROUP_9:th.97">Th97</a></i>;<br/><a NAME="E6:205_1_2"/><b>then </b><i><font color="Green" title="E183">A12</font></i>: 
 ex <font color="Olive" title="b1">p'</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <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="c5">s1'</font></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">s1'</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c6">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>
 <b>by </b><i><a class="txt" href="group_9.html#E4:205_1_2"><i><font color="Green" title="E181">A10</font></i></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>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 not <font color="Olive" title="b1">s1'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Olive" title="b2">s2'</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">s1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1 &amp; <font color="Olive" title="b1">s1'</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">s2'</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  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b1">s1'</font></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b1">s1'</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b2">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> 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  <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> 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/><a NAME="E7:205_1_2"/><i><font color="Green" title="E184">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="205_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s1'</font>, <font color="Maroon" title="c8">s2'</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>;<br/>



<b>assume </b><a NAME="E1:205_1_2_2"/><i><font color="Green" title="E185">A14</font></i>: 
( not <font color="Maroon" title="c7">s1'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c8">s2'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/>

<b>assume </b><a NAME="E2:205_1_2_2"/><i><font color="Green" title="E186">A15</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">s1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 &amp; <font color="Maroon" title="c7">s1'</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> )
 ;<br/>

<b>assume </b><a NAME="E3:205_1_2_2"/><i><font color="Green" title="E187">A16</font></i>: 
<font color="Maroon" title="c8">s2'</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c9">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <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">s1'</font></span>)</span><b> such that </b><a NAME="E5:205_1_2_2"/><i><font color="Green" title="E188">A17</font></i>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c7">s1'</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c8">s2'</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c1">O</font>
 ;<br/>

<a NAME="E6:205_1_2_2"/><i><font color="Green" title="E189">A18</font></i>: 
<font color="Maroon" title="c7">s1'</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c8">s2'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:205_1_2_2"><i><font color="Green" title="E185">A14</font></i></a>, <a class="txt" href="group_9.html#E5:205_1_2_2"><i><font color="Green" title="E188">A17</font></i></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</a></i>;<br/>
<a NAME="E7:205_1_2_2"/>
<font color="Maroon" title="c7">s1'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E2:205_1_2_2"><i><font color="Green" title="E186">A15</font></i></a>, <a class="ref" href="group_9.html#T96" target="_self" title="GROUP_9:th.96">Th96</a></i>;<br/>
<a NAME="E8:205_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">s2'</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:205"><i><font color="Green" title="E177">A1</font></i></a>, <a class="txt" href="group_9.html#E6:205_1_2_2"><i><font color="Green" title="E189">A18</font></i></a>, <a class="ref" href="group_9.html#T115" target="_self" title="GROUP_9:th.115">Th115</a></i>;<br/>
<a NAME="E9:205_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">s2'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">s2</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E3:205_1_2_2"><i><font color="Green" title="E187">A16</font></i></a>, <a class="txt" href="group_9.html#E2:205"><i><font color="Green" title="E178">A2</font></i></a>, <a class="ref" href="group_9.html#T98" target="_self" title="GROUP_9:th.98">Th98</a></i>;<br/>
<a NAME="E10:205_1_2_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="group_9.html#E6:205_1_2_2"><i><font color="Green" title="E189">A18</font></i></a>, <a class="txt" href="group_9.html#E2:205_1_2_2"><i><font color="Green" title="E186">A15</font></i></a>, <a class="ref" href="group_9.html#T96" target="_self" title="GROUP_9:th.96">Th96</a></i>;<br/>
<b>hence </b><a NAME="E11:205_1_2_2"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <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> 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>
 <b>by </b><i><a class="txt" href="group_9.html#E1:205_1_2"><i><font color="Green" title="E179">A6</font></i></a>, <a class="txt" href="group_9.html#E2:205_1_2"><i><font color="Green" title="E180">A7</font></i></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:205_1_2"/><i><font color="Green" title="E185">A19</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="205_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:205_1_2_3"/><i><font color="Green" title="E186">A20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="205_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">s1'</font>, <font color="Maroon" title="c9">s2'</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>;<br/><b>assume </b><a NAME="E1:205_1_2_3_1"/>
( not <font color="Maroon" title="c8">s1'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c9">s2'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/><b>assume </b><a NAME="E2:205_1_2_3_1"/><i><font color="Green" title="E187">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s1'</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><b>assume </b><a NAME="E3:205_1_2_3_1"/><i><font color="Green" title="E188">A22</font></i>: 
<font color="Maroon" title="c8">s1'</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font>
 ;<br/><b>assume </b><a NAME="E4:205_1_2_3_1"/><i><font color="Green" title="E189">A23</font></i>: 
<font color="Maroon" title="c9">s2'</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font>
 ;<br/><b>set </b><font color="Maroon" title="c10">f1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c8">s1'</font>;<br/><b>set </b><font color="Maroon" title="c11">f2</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c9">s2'</font>;<br/><b>given </b><font color="Maroon" title="c12">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <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="c8">s1'</font></span>)</span><b> such that </b><a NAME="E6:205_1_2_3_1"/><i><font color="Green" title="E190">A24</font></i>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c8">s1'</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c9">s2'</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c1">O</font>
 ;<br/><a NAME="E7:205_1_2_3_1"/><i><font color="Green" title="E191">A25</font></i>: 
 <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="c8">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="c9">s2'</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E6:205_1_2_3_1"><i><font color="Green" title="E190">A24</font></i></a>, <a class="ref" href="group_9.html#D37" target="_self" title="GROUP_9:def.37">Def37</a></i>;<br/><a NAME="E8:205_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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> 0 <a href="nat_1.html#K1" title="NAT_1:func.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>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">i</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E10:205_1_2_3_1"/><i><font color="Green" title="E192">A26</font></i>: 
( <font color="Maroon" title="c13">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="c8">s1'</font></span>)</span> &amp; ( for <font color="Olive" title="b1">H</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c8">s1'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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> ) )
 <b>by </b><i><a class="txt" href="group_9.html#E2:205_1_2_3_1"><i><font color="Green" title="E187">A21</font></i></a>, <a class="ref" href="group_9.html#T109" target="_self" title="GROUP_9:th.109">Th109</a>, <a class="txt" href="group_9.html#E3:205_1_2_3_1"><i><font color="Green" title="E188">A22</font></i></a>, <a class="txt" href="group_9.html#E1:205"><i><font color="Green" title="E177">A1</font></i></a></i>;<br/><a NAME="E11:205_1_2_3_1"/><i><font color="Green" title="E193">A27</font></i>: 
( <font color="Maroon" title="c13">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="c8">s1'</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c8">s1'</font> &amp; <font color="Maroon" title="c8">s1'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E10:205_1_2_3_1"><i><font color="Green" title="E192">A26</font></i></a>, <a class="ref" href="group_9.html#T103" target="_self" title="GROUP_9:th.103">Th103</a></i>;<br/><b>set </b><font color="Maroon" title="c14">j</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c13">i</font>;<br/><a NAME="E12:205_1_2_3_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c8">s1'</font></span>)</span>
 ;<br/><a NAME="E13:205_1_2_3_1"/><b>then </b><i><font color="Green" title="E194">A28</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c8">s1'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:205_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E10:205_1_2_3_1"><i><font color="Green" title="E192">A26</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/><a NAME="E15:205_1_2_3_1"/><b>then </b><i><font color="Green" title="E195">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c13">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="c8">s1'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E13:205_1_2_3_1"><i><font color="Green" title="E194">A28</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">j</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c13">i</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E17:205_1_2_3_1"/><i><font color="Green" title="E196">A30</font></i>: 
<font color="Maroon" title="c15">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="c9">s2'</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E15:205_1_2_3_1"><i><font color="Green" title="E195">A29</font></i></a>, <a class="txt" href="group_9.html#E7:205_1_2_3_1"><i><font color="Green" title="E191">A25</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c16">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="c8">s1'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>;<br/><b>set </b><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="c9">s2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>;<br/><a NAME="E18:205_1_2_3_1"/><i><font color="Green" title="E197">A31</font></i>: 
<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="c8">s1'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="c8">s1'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="txt" href="group_9.html#E10:205_1_2_3_1"><i><font color="Green" title="E192">A26</font></i></a></i>;<br/><a NAME="E19:205_1_2_3_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="c9">s2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c9">s2'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="txt" href="group_9.html#E17:205_1_2_3_1"><i><font color="Green" title="E196">A30</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">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="c8">s1'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c19">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="c9">s2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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> <b>by </b><i><a class="txt" href="group_9.html#E18:205_1_2_3_1"><i><font color="Green" title="E197">A31</font></i></a>, <a class="ref" href="group_9.html#T102" target="_self" title="GROUP_9:th.102">Th102</a></i>;<br/><a NAME="E21:205_1_2_3_1"/><i><font color="Green" title="E198">A32</font></i>: 
<font color="Maroon" title="c18">H1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E10:205_1_2_3_1"><i><font color="Green" title="E192">A26</font></i></a></i>;<br/><a NAME="E22:205_1_2_3_1"/>
<font color="Maroon" title="c18">H1</font>,<font color="Maroon" title="c19">H2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E6:205_1_2_3_1"><i><font color="Green" title="E190">A24</font></i></a>, <a class="txt" href="group_9.html#E10:205_1_2_3_1"><i><font color="Green" title="E192">A26</font></i></a>, <a class="ref" href="group_9.html#D37" target="_self" title="GROUP_9:def.37">Def37</a></i>;<br/><a NAME="E23:205_1_2_3_1"/><b>then </b>
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="c9">s2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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>
 <b>by </b><i><a class="ref" href="group_9.html#T58" target="_self" title="GROUP_9:th.58">Th58</a>, <a class="txt" href="group_9.html#E21:205_1_2_3_1"><i><font color="Green" title="E198">A32</font></i></a></i>;<br/><a NAME="E24:205_1_2_3_1"/><b>then </b><i><font color="Green" title="E199">A33</font></i>: 
( <font color="Maroon" title="c15">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="c9">s2'</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c9">s2'</font> &amp; <font color="Maroon" title="c9">s2'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s2'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E17:205_1_2_3_1"><i><font color="Green" title="E196">A30</font></i></a>, <a class="ref" href="group_9.html#T103" target="_self" title="GROUP_9:th.103">Th103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">s1''</font> =  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c8">s1'</font>,<font color="Maroon" title="c13">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> <b>by </b><i><a class="ref" href="finseq_3.html#T114" title="FINSEQ_3:th.114">FINSEQ_3:114</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">s1''</font> = <font color="Maroon" title="c20">s1''</font> as   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T94" target="_self" title="GROUP_9:th.94">Th94</a>, <a class="txt" href="group_9.html#E11:205_1_2_3_1"><i><font color="Green" title="E193">A27</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">s2''</font> =  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c9">s2'</font>,<font color="Maroon" title="c15">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> <b>by </b><i><a class="ref" href="finseq_3.html#T114" title="FINSEQ_3:th.114">FINSEQ_3:114</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">s2''</font> = <font color="Maroon" title="c22">s2''</font> as   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T94" target="_self" title="GROUP_9:th.94">Th94</a>, <a class="txt" href="group_9.html#E24:205_1_2_3_1"><i><font color="Green" title="E199">A33</font></i></a></i>;<br/><a NAME="E29:205_1_2_3_1"/><i><font color="Green" title="E200">A34</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">s1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T118" title="FINSEQ_3:th.118">FINSEQ_3:118</a>, <a class="txt" href="group_9.html#E2:205_1_2_3_1"><i><font color="Green" title="E187">A21</font></i></a>, <a class="txt" href="group_9.html#E11:205_1_2_3_1"><i><font color="Green" title="E193">A27</font></i></a></i>;<br/><a NAME="E30:205_1_2_3_1"/><i><font color="Green" title="E201">A35</font></i>: 
<font color="Maroon" title="c21">s1''</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="ref" href="group_9.html#T99" target="_self" title="GROUP_9:th.99">Th99</a>, <a class="txt" href="group_9.html#E11:205_1_2_3_1"><i><font color="Green" title="E193">A27</font></i></a>, <a class="txt" href="group_9.html#E3:205_1_2_3_1"><i><font color="Green" title="E188">A22</font></i></a>, <a class="txt" href="group_9.html#E1:205"><i><font color="Green" title="E177">A1</font></i></a></i>;<br/><a NAME="E31:205_1_2_3_1"/><i><font color="Green" title="E202">A36</font></i>: 
<font color="Maroon" title="c23">s2''</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c4">s2</font>
 <b>by </b><i><a class="ref" href="group_9.html#T99" target="_self" title="GROUP_9:th.99">Th99</a>, <a class="txt" href="group_9.html#E24:205_1_2_3_1"><i><font color="Green" title="E199">A33</font></i></a>, <a class="txt" href="group_9.html#E4:205_1_2_3_1"><i><font color="Green" title="E189">A23</font></i></a>, <a class="txt" href="group_9.html#E2:205"><i><font color="Green" title="E178">A2</font></i></a></i>;<br/><a NAME="E32:205_1_2_3_1"/><i><font color="Green" title="E203">A37</font></i>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c21">s1''</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="c8">s1'</font></span>)</span>,<font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="ref" href="group_9.html#T104" target="_self" title="GROUP_9:th.104">Th104</a>, <a class="txt" href="group_9.html#E11:205_1_2_3_1"><i><font color="Green" title="E193">A27</font></i></a></i>;<br/><a NAME="E33:205_1_2_3_1"/><i><font color="Green" title="E204">A38</font></i>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c23">s2''</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="c9">s2'</font></span>)</span>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="ref" href="group_9.html#T104" target="_self" title="GROUP_9:th.104">Th104</a>, <a class="txt" href="group_9.html#E24:205_1_2_3_1"><i><font color="Green" title="E199">A33</font></i></a></i>;<br/><a NAME="E34:205_1_2_3_1"/><i><font color="Green" title="E205">A39</font></i>: 
( not <font color="Maroon" title="c21">s1''</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c23">s2''</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="group_9.html#E30:205_1_2_3_1"><i><font color="Green" title="E201">A35</font></i></a>, <a class="txt" href="group_9.html#E31:205_1_2_3_1"><i><font color="Green" title="E202">A36</font></i></a>, <a class="txt" href="group_9.html#E1:205_1_2"><i><font color="Green" title="E179">A6</font></i></a>, <a class="txt" href="group_9.html#E2:205_1_2"><i><font color="Green" title="E180">A7</font></i></a>, <a class="ref" href="group_9.html#T97" target="_self" title="GROUP_9:th.97">Th97</a></i>;<br/><a NAME="E35:205_1_2_3_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <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="c21">s1''</font></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c21">s1''</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c23">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>
 <b>by </b><i><a class="txt" href="group_9.html#E6:205_1_2_3_1"><i><font color="Green" title="E190">A24</font></i></a>, <a class="txt" href="group_9.html#E10:205_1_2_3_1"><i><font color="Green" title="E192">A26</font></i></a>, <a class="txt" href="group_9.html#E32:205_1_2_3_1"><i><font color="Green" title="E203">A37</font></i></a>, <a class="txt" href="group_9.html#E33:205_1_2_3_1"><i><font color="Green" title="E204">A38</font></i></a>, <a class="ref" href="group_9.html#T106" target="_self" title="GROUP_9:th.106">Th106</a></i>;<br/><b>hence </b><a NAME="E36:205_1_2_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="group_9.html#E34:205_1_2_3_1"><i><font color="Green" title="E205">A39</font></i></a>, <a class="txt" href="group_9.html#E29:205_1_2_3_1"><i><font color="Green" title="E200">A34</font></i></a>, <a class="txt" href="group_9.html#E30:205_1_2_3_1"><i><font color="Green" title="E201">A35</font></i></a>, <a class="txt" href="group_9.html#E31:205_1_2_3_1"><i><font color="Green" title="E202">A36</font></i></a>, <a class="txt" href="group_9.html#E1:205_1_2_3"><i><font color="Green" title="E186">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:205_1_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E9:205_1_2"/><i><font color="Green" title="E186">A40</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="group_9.html#E7:205_1_2"><i><font color="Green" title="E184">A13</font></i></a>, <a class="txt" href="group_9.html#E8:205_1_2"><i><font color="Green" title="E185">A19</font></i></a>);<br/></i><a NAME="E10:205_1_2"/>
 ex <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:205_1_2"><i><font color="Green" title="E181">A10</font></i></a>, <a class="ref" href="group_9.html#T95" target="_self" title="GROUP_9:th.95">Th95</a></i>;<br/><a NAME="E11:205_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <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> 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>
 <b>by </b><i><a class="txt" href="group_9.html#E5:205_1_2"><i><font color="Green" title="E182">A11</font></i></a>, <a class="txt" href="group_9.html#E4:205_1_2"><i><font color="Green" title="E181">A10</font></i></a>, <a class="txt" href="group_9.html#E6:205_1_2"><i><font color="Green" title="E183">A12</font></i></a>, <a class="txt" href="group_9.html#E9:205_1_2"><i><font color="Green" title="E186">A40</font></i></a></i>;<br/><b>hence </b><a NAME="E12:205_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>
 <b>by </b><i><a class="txt" href="group_9.html#E1:205_1_2"><i><font color="Green" title="E179">A6</font></i></a>, <a class="txt" href="group_9.html#E2:205_1_2"><i><font color="Green" title="E180">A7</font></i></a>, <a class="ref" href="group_9.html#T108" target="_self" title="GROUP_9:th.108">Th108</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
