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<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/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="204_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:204_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#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <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 ) 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#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or  <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#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="204_1_1"><b>suppose </b></a><a NAME="E1:204_1_1"/><i><font color="Green" title="E176">A1</font></i>: 
(  <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 &amp;  <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 )
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c5">s1'</font> =  <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>;<br/><b>set </b><font color="Maroon" title="c6">s2'</font> =  <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c4">s2</font>,<font color="Maroon" title="c3">s1</font>;<br/><b>take </b>
 <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>
;<br/><b>take </b>
 <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c4">s2</font>,<font color="Maroon" title="c3">s1</font>
;<br/><b>thus </b><a NAME="E2:204_1_1"/>
(  <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c3">s1</font>,<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> &amp;  <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c4">s2</font>,<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="txt" href="group_9.html#E1:204_1_1"><i><font color="Green" title="E176">A1</font></i></a>, <a class="ref" href="group_9.html#T116" target="_self" title="GROUP_9:th.116">Th116</a></i>;<br/><b>thus </b><a NAME="E3:204_1_1"/>
 <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a>  <a href="group_9.html#K21" title="GROUP_9:func.21">the_schreier_series_of</a> <font color="Maroon" title="c4">s2</font>,<font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:204_1_1"><i><font color="Green" title="E176">A1</font></i></a>, <a class="ref" href="group_9.html#T117" target="_self" title="GROUP_9:th.117">Th117</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="204_1_2"><b>suppose </b></a><a NAME="E1:204_1_2"/><i><font color="Green" title="E176">A2</font></i>: 
(  <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> 1 or  <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#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="204_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:204_1_2_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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> 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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="204_1_2_1_1"><b>suppose </b></a><a NAME="E1:204_1_2_1_1"/><i><font color="Green" title="E177">A3</font></i>: 
 <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c5">s1'</font> = <font color="Maroon" title="c4">s2</font>;<br/><b>set </b><font color="Maroon" title="c6">s2'</font> = <font color="Maroon" title="c4">s2</font>;<br/><b>take </b>
<font color="Maroon" title="c4">s2</font>
;<br/><b>take </b>
<font color="Maroon" title="c4">s2</font>
;<br/><b>thus </b><a NAME="E2:204_1_2_1_1"/>
( <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> &amp; <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="c4">s2</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:204_1_2"><i><font color="Green" title="E176">A2</font></i></a>, <a class="txt" href="group_9.html#E1:204_1_2_1_1"><i><font color="Green" title="E177">A3</font></i></a>, <a class="ref" href="group_9.html#T114" target="_self" title="GROUP_9:th.114">Th114</a></i>;<br/><b>thus </b><a NAME="E3:204_1_2_1_1"/>
<font color="Maroon" title="c4">s2</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#T113" target="_self" title="GROUP_9:th.113">Th113</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="204_1_2_1_2"><b>suppose </b></a><a NAME="E1:204_1_2_1_2"/><i><font color="Green" title="E177">A4</font></i>: 
 <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c5">s1'</font> = <font color="Maroon" title="c3">s1</font>;<br/><b>set </b><font color="Maroon" title="c6">s2'</font> = <font color="Maroon" title="c3">s1</font>;<br/><b>take </b>
<font color="Maroon" title="c3">s1</font>
;<br/><b>take </b>
<font color="Maroon" title="c3">s1</font>
;<br/><b>thus </b><a NAME="E2:204_1_2_1_2"/>
( <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="c3">s1</font> &amp; <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="txt" href="group_9.html#E1:204_1_2"><i><font color="Green" title="E176">A2</font></i></a>, <a class="txt" href="group_9.html#E1:204_1_2_1_2"><i><font color="Green" title="E177">A4</font></i></a>, <a class="ref" href="group_9.html#T114" target="_self" title="GROUP_9:th.114">Th114</a></i>;<br/><b>thus </b><a NAME="E3:204_1_2_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="c3">s1</font>
 <b>by </b><i><a class="ref" href="group_9.html#T113" target="_self" title="GROUP_9:th.113">Th113</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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