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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/>  for <font color="Olive" title="b3">s1</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Olive" title="b1">G1</font><br/>  for <font color="Olive" title="b4">s2</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Olive" title="b2">G2</font>  st  not <font color="Olive" title="b3">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  not <font color="Olive" title="b4">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/>( <font color="Olive" title="b3">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Olive" title="b4">s2</font> iff  ex <font color="Olive" title="b5">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b3">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b3">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b4">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b5">p</font>,<font color="Maroon" title="c1">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G1</font>, <font color="Maroon" title="c3">G2</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s1</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b2">s2</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c3">G2</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>  holds <br/>( <font color="Olive" title="b1">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Olive" title="b2">s2</font> iff  ex <font color="Olive" title="b3">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b1">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s1</font> be   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c2">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s2</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c3">G2</font>  st  not <font color="Maroon" title="c4">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  not <font color="Olive" title="b1">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/>( <font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Olive" title="b1">s2</font> iff  ex <font color="Olive" title="b2">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="c4">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Olive" title="b1">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">s2</font> be   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c3">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  not <font color="Maroon" title="c5">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies ( <font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font> iff  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:180"/><span class="lab"><font color="Green" title="E154">A1</font></span>: 
 not <font color="Maroon" title="c4">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E2:180"/><span class="lab"><font color="Green" title="E155">A2</font></span>: 
 not <font color="Maroon" title="c5">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font> iff  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">f2</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">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> implies <font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:180_1"/><span class="lab"><font color="Green" title="E156">A3</font></span>: 
<font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font></span><br/><a NAME="E2:180_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def35</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_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="E1:180_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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">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="180_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:180_1_1_1"/><span class="lab"><font color="Green" title="E158">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">O</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">fs1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c9">fs2</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c10">p</font> =  the   <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="c4">s1</font></span>)</span></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">pf</font> =  the   <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="c4">s1</font></span>)</span></span>)</span> as   <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> <font color="Maroon" title="c8">fs1</font></span>)</span> ;<br/><a NAME="E4:180_1_1_1"/>
<font color="Maroon" title="c8">fs1</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="#E1:180_1_1_1"><span class="lab"><font color="Green" title="E158">A5</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><a NAME="E5:180_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A6</font></span>: 
 for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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">fs1</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">pf</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="Olive" title="b3">i</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> holds <br/><font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 ;<br/><span class="kw">take </span><font color="Maroon" title="c12">p</font> =  the   <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="c4">s1</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">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></span><br/><a NAME="E6:180_1_1_1"/>
<font color="Maroon" title="c9">fs2</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="#E2:180_1"><span class="lab"><font color="Green" title="E157">A4</font></span></a>, <a class="txt" href="#E1:180_1_1_1"><span class="lab"><font color="Green" title="E158">A5</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><a NAME="E7:180_1_1_1"/><span class="kw">then </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="c4">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="c5">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_1_1_1"><span class="lab"><font color="Green" title="E158">A5</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><span class="kw">hence </span><a NAME="E8:180_1_1_1"/>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:180_1_1_1"><span class="lab"><font color="Green" title="E159">A6</font></span></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="180_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:180_1_1_2"/><span class="lab"><font color="Green" title="E158">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p9</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> st  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Olive" title="b1">p9</font>,<font color="Maroon" title="c1">O</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">n</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><a NAME="E2:180_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_1_1_2"><span class="lab"><font color="Green" title="E158">A7</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:180_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">n</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E5:180_1_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:180_1_1_2"/><span class="lab"><font color="Green" title="E159">A8</font></span>: 
 for <font color="Olive" title="b1">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b2">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b3">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H1</font><br/>  for <font color="Olive" title="b4">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H2</font><br/>  for <font color="Olive" title="b5">i</font>, <font color="Olive" title="b6">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">j</font> &amp; <font color="Olive" title="b3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N1</font>,<font color="Olive" title="b2">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">N2</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="#E1:180_1"><span class="lab"><font color="Green" title="E156">A3</font></span></a>, <a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def35</a></span>;<br/><a NAME="E8:180_1_1_2"/><span class="lab"><font color="Green" title="E160">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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> <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">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_1_1_2"><span class="lab"><font color="Green" title="E158">A7</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><a NAME="E9:180_1_1_2"/><span class="kw">then </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="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font>
 <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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">p9</font> = <font color="Maroon" title="c10">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  as   <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="c4">s1</font></span>)</span></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">fs1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c13">fs2</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/><a NAME="E12:180_1_1_2"/><span class="lab"><font color="Green" title="E161">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s2</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> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_1"><span class="lab"><font color="Green" title="E157">A4</font></span></a>, <a class="txt" href="#E1:180_1_1_2"><span class="lab"><font color="Green" title="E158">A7</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">pf</font> = <font color="Maroon" title="c11">p9</font> as   <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> <font color="Maroon" title="c12">fs1</font></span>)</span> ;<br/><span class="kw">take </span><font color="Maroon" title="c15">p9</font> = <font color="Maroon" title="c11">p9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">O</font></span><br/><a NAME="E14:180_1_1_2"/><span class="lab"><font color="Green" title="E162">A11</font></span>: 
<font color="Maroon" title="c14">pf</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T43" title="FUNCT_1:th.43">FUNCT_1:43</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">H19</font>, <font color="Maroon" title="c17">H29</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c12">fs1</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pf</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="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c16">H19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c17">H29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c16">H19</font>,<font color="Maroon" title="c17">H29</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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="c12">fs1</font> &amp; <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pf</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="c18">i</font> &amp; <font color="Maroon" title="c16">H19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c17">H29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> implies <font color="Maroon" title="c16">H19</font>,<font color="Maroon" title="c17">H29</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">set </span><font color="Maroon" title="c20">H1</font> = <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c21">H2</font> = <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>;<br/><span class="kw">set </span><font color="Maroon" title="c22">N1</font> = <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">N2</font> = <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:180_1_1_2_1"/><span class="lab"><font color="Green" title="E163">A12</font></span>: 
<font color="Maroon" title="c18">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="c12">fs1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pf</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="c18">i</font> &amp; <font color="Maroon" title="c16">H19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c17">H29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> implies <font color="Maroon" title="c16">H19</font>,<font color="Maroon" title="c17">H29</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><a NAME="E2:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">A13</font></span>: 
<font color="Maroon" title="c18">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> <font color="Maroon" title="c12">fs1</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="E3:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:180_1_1_2_1"/><span class="lab"><font color="Green" title="E166">A15</font></span>: 
<font color="Maroon" title="c18">i</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="c12">fs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_1_1_2_1"><span class="lab"><font color="Green" title="E164">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E167">A16</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">fs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:180_1_1_2_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:180_1_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:180_1_1_2_1"><span class="lab"><font color="Green" title="E165">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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> <font color="Maroon" title="c4">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:180_1_1_2"><span class="lab"><font color="Green" title="E160">A9</font></span></a>, <a class="txt" href="#E5:180_1_1_2_1"><span class="lab"><font color="Green" title="E167">A16</font></span></a></span>;<br/><a NAME="E9:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A17</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">s1</font>
 <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/><span class="kw">assume </span><a NAME="E10:180_1_1_2_1"/><span class="lab"><font color="Green" title="E169">A18</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pf</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="c18">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">H19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c17">H29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> implies <font color="Maroon" title="c16">H19</font>,<font color="Maroon" title="c17">H29</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><a NAME="E11:180_1_1_2_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">fs1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">fs1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E12:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</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">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:180_1_1_2"><span class="lab"><font color="Green" title="E160">A9</font></span></a>, <a class="txt" href="#E4:180_1_1_2_1"><span class="lab"><font color="Green" title="E166">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">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> <font color="Maroon" title="c4">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:180_1_1_2_1"><span class="lab"><font color="Green" title="E165">A14</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E14:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A19</font></span>: 
<font color="Maroon" title="c18">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="c4">s1</font>
 <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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">H1</font> = <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c25">N1</font> = <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:180_1_1_2_1"><span class="lab"><font color="Green" title="E168">A17</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">H1</font> = <font color="Maroon" title="c24">H1</font>, <font color="Maroon" title="c27">N1</font> = <font color="Maroon" title="c25">N1</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">N1</font> = <font color="Maroon" title="c27">N1</font> as   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c26">H1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:180_1_1_2_1"><span class="lab"><font color="Green" title="E170">A19</font></span></a>, <a class="txt" href="#E9:180_1_1_2_1"><span class="lab"><font color="Green" title="E168">A17</font></span></a>, <a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E18:180_1_1_2_1"/><span class="lab"><font color="Green" title="E171">A20</font></span>: 
<font color="Maroon" title="c16">H19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">and </span><br/><a NAME="E19:180_1_1_2_1"/><span class="lab"><font color="Green" title="E172">A21</font></span>: 
<font color="Maroon" title="c17">H29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">H19</font>,<font color="Maroon" title="c17">H29</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><a NAME="E20:180_1_1_2_1"/>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:180_1_1_2"><span class="lab"><font color="Green" title="E160">A9</font></span></a>, <a class="txt" href="#E2:180_1_1_2_1"><span class="lab"><font color="Green" title="E164">A13</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E21:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A22</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:180_1_1_2"><span class="lab"><font color="Green" title="E162">A11</font></span></a>, <a class="txt" href="#E10:180_1_1_2_1"><span class="lab"><font color="Green" title="E169">A18</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E22:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A23</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E23:180_1_1_2_1"/><span class="lab"><font color="Green" title="E175">A24</font></span>: 
<font color="Maroon" title="c19">j</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="c13">fs2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_1"><span class="lab"><font color="Green" title="E157">A4</font></span></a>, <a class="txt" href="#E12:180_1_1_2"><span class="lab"><font color="Green" title="E161">A10</font></span></a>, <a class="txt" href="#E21:180_1_1_2_1"><span class="lab"><font color="Green" title="E173">A22</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E24:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A25</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">fs2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E25:180_1_1_2_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E26:180_1_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:180_1_1_2_1"><span class="lab"><font color="Green" title="E174">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E27:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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> <font color="Maroon" title="c5">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:180_1_1_2"><span class="lab"><font color="Green" title="E161">A10</font></span></a>, <a class="txt" href="#E24:180_1_1_2_1"><span class="lab"><font color="Green" title="E176">A25</font></span></a></span>;<br/><a NAME="E28:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A26</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">s2</font>
 <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="E29:180_1_1_2_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">fs2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">fs2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E30:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">j</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="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:180_1_1_2"><span class="lab"><font color="Green" title="E161">A10</font></span></a>, <a class="txt" href="#E23:180_1_1_2_1"><span class="lab"><font color="Green" title="E175">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E31:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c5">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:180_1_1_2_1"><span class="lab"><font color="Green" title="E174">A23</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E32:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A27</font></span>: 
<font color="Maroon" title="c19">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="c5">s2</font>
 <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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">H2</font> = <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>, <font color="Maroon" title="c30">N2</font> = <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c3">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:180_1_1_2_1"><span class="lab"><font color="Green" title="E177">A26</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">H2</font> = <font color="Maroon" title="c29">H2</font>, <font color="Maroon" title="c32">N2</font> = <font color="Maroon" title="c30">N2</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c33">N2</font> = <font color="Maroon" title="c32">N2</font> as   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c31">H2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:180_1_1_2_1"><span class="lab"><font color="Green" title="E178">A27</font></span></a>, <a class="txt" href="#E28:180_1_1_2_1"><span class="lab"><font color="Green" title="E177">A26</font></span></a>, <a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/><a NAME="E36:180_1_1_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:180_1_1_2"><span class="lab"><font color="Green" title="E160">A9</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="E37:180_1_1_2_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_1_1_2_1"><span class="lab"><font color="Green" title="E163">A12</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E38:180_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E179">A28</font></span>: 
<font color="Maroon" title="c26">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c28">N1</font>,<font color="Maroon" title="c31">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c33">N2</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="#E7:180_1_1_2"><span class="lab"><font color="Green" title="E159">A8</font></span></a>, <a class="txt" href="#E14:180_1_1_2"><span class="lab"><font color="Green" title="E162">A11</font></span></a>, <a class="txt" href="#E10:180_1_1_2_1"><span class="lab"><font color="Green" title="E169">A18</font></span></a></span>;<br/><a NAME="E39:180_1_1_2_1"/>
<font color="Maroon" title="c19">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="c5">s2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_1"><span class="lab"><font color="Green" title="E157">A4</font></span></a>, <a class="txt" href="#E12:180_1_1_2"><span class="lab"><font color="Green" title="E161">A10</font></span></a>, <a class="txt" href="#E21:180_1_1_2_1"><span class="lab"><font color="Green" title="E173">A22</font></span></a></span>;<br/><a NAME="E40:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">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="c13">fs2</font>
 <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="E41:180_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c33">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">H29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_1"><span class="lab"><font color="Green" title="E157">A4</font></span></a>, <a class="txt" href="#E1:180_1_1_2"><span class="lab"><font color="Green" title="E158">A7</font></span></a>, <a class="txt" href="#E19:180_1_1_2_1"><span class="lab"><font color="Green" title="E172">A21</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><span class="kw">hence </span><a NAME="E42:180_1_1_2_1"/>
<font color="Maroon" title="c16">H19</font>,<font color="Maroon" title="c17">H29</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="#E1:180_1_1_2"><span class="lab"><font color="Green" title="E158">A7</font></span></a>, <a class="txt" href="#E1:180_1_1_2_1"><span class="lab"><font color="Green" title="E163">A12</font></span></a>, <a class="txt" href="#E18:180_1_1_2_1"><span class="lab"><font color="Green" title="E171">A20</font></span></a>, <a class="txt" href="#E38:180_1_1_2_1"><span class="lab"><font color="Green" title="E179">A28</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E16:180_1_1_2"/>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_1"><span class="lab"><font color="Green" title="E157">A4</font></span></a>, <a class="txt" href="#E8:180_1_1_2"><span class="lab"><font color="Green" title="E160">A9</font></span></a>, <a class="txt" href="#E12:180_1_1_2"><span class="lab"><font color="Green" title="E161">A10</font></span></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>

<span class="kw">given </span><font color="Maroon" title="c8">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="c4">s1</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E5:180"/><span class="lab"><font color="Green" title="E156">A29</font></span>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s1</font>, <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</font> <a href="group_9.html#R6" title="GROUP_9:pred.6">are_equivalent_under</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font></span><br/>

<a NAME="E6:180"/><span class="lab"><font color="Green" title="E157">A30</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="c4">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="c5">s2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:180"><span class="lab"><font color="Green" title="E156">A29</font></span></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def37</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_2"><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="E1:180_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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">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="180_2_1"><span class="kw">suppose </span></a><a NAME="E1:180_2_1"/><span class="lab"><font color="Green" title="E158">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font></span><br/><div class="add"><a NAME="E2:180_2_1"/><span class="lab"><font color="Green" title="E159">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180"><span class="lab"><font color="Green" title="E154">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><span class="lab"><font color="Green" title="E160">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="180_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> implies  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> st <br/> for <font color="Olive" title="b2">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b3">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b4">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H1</font><br/>  for <font color="Olive" title="b5">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">H2</font><br/>  for <font color="Olive" title="b6">i</font>, <font color="Olive" title="b7">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">j</font> &amp; <font color="Olive" title="b4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b5">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">N1</font>,<font color="Olive" title="b3">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b5">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">set </span><font color="Maroon" title="c10">p</font> =  the   <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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:180_2_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> st <br/> for <font color="Olive" title="b2">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b3">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b4">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H1</font><br/>  for <font color="Olive" title="b5">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">H2</font><br/>  for <font color="Olive" title="b6">i</font>, <font color="Olive" title="b7">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">j</font> &amp; <font color="Olive" title="b4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b5">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">N1</font>,<font color="Olive" title="b3">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b5">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><a NAME="E2:180_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_2_1"><span class="lab"><font color="Green" title="E158">A31</font></span></a>, <a class="txt" href="#E2:180_2_1"><span class="lab"><font color="Green" title="E159">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:180_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A34</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">take </span><font color="Maroon" title="c11">p</font> =  the   <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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b2">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b3">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H1</font><br/>  for <font color="Olive" title="b4">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H2</font><br/>  for <font color="Olive" title="b5">i</font>, <font color="Olive" title="b6">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">j</font> &amp; <font color="Olive" title="b3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N1</font>,<font color="Olive" title="b2">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c12">H1</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b2">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c12">H1</font><br/>  for <font color="Olive" title="b3">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H2</font><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b1">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b2">N1</font>,<font color="Olive" title="b1">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c13">H2</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c12">H1</font><br/>  for <font color="Olive" title="b2">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c13">H2</font><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b2">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c14">N1</font> be   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c12">H1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c13">H2</font><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c15">N2</font> be   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c13">H2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c16">i</font>, <font color="Maroon" title="c17">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:180_2_1_1"/><span class="lab"><font color="Green" title="E162">A35</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">and </span><br/><a NAME="E5:180_2_1_1"/>
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:180_2_1_1"/>
<font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">and </span><br/><a NAME="E7:180_2_1_1"/>
<font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:180_2_1_1"/>
<font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:180_2_1_1"/>
<font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">thus </span><a NAME="E10:180_2_1_1"/>
<font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</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="#E3:180_2_1_1"><span class="lab"><font color="Green" title="E161">A34</font></span></a>, <a class="txt" href="#E4:180_2_1_1"><span class="lab"><font color="Green" title="E162">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:180_2_1"/><span class="lab"><font color="Green" title="E161">A36</font></span>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">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="#E1:180_2_1"><span class="lab"><font color="Green" title="E158">A31</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:180_2_1_2"/><span class="lab"><font color="Green" title="E162">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:180_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180"><span class="lab"><font color="Green" title="E155">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:180_2_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="#E1:180_2_1_2"><span class="lab"><font color="Green" title="E162">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:180_2_1_2"/><span class="kw">then </span>
<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="c5">s2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><span class="kw">hence </span><a NAME="E5:180_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:180"><span class="lab"><font color="Green" title="E157">A30</font></span></a>, <a class="txt" href="#E4:180_2_1"><span class="lab"><font color="Green" title="E161">A36</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:180_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_2_1"><span class="lab"><font color="Green" title="E158">A31</font></span></a>, <a class="txt" href="#E2:180_2_1"><span class="lab"><font color="Green" title="E159">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:180_2_1"/>
<font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:180_2_1"><span class="lab"><font color="Green" title="E160">A33</font></span></a>, <a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="180_2_2"><span class="kw">suppose </span></a><a NAME="E1:180_2_2"/><span class="lab"><font color="Green" title="E158">A38</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font></span><br/><div class="add"><a NAME="E2:180_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A39</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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> <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">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><div><span class="lab"><font color="Green" title="E160">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="180_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:180_2_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:180_2_2_1"/><span class="kw">then </span>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c5">s2</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="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><a NAME="E3:180_2_2_1"/><span class="kw">then </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="c5">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E4:180_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:180"><span class="lab"><font color="Green" title="E157">A30</font></span></a>, <a class="txt" href="#E1:180_2_2"><span class="lab"><font color="Green" title="E158">A38</font></span></a>, <a class="txt" href="#E2:180_2_2"><span class="lab"><font color="Green" title="E159">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E161">A41</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="180_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> implies  ex <font color="Olive" title="b1">p9</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> st <br/> for <font color="Olive" title="b2">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b3">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b4">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H1</font><br/>  for <font color="Olive" title="b5">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">H2</font><br/>  for <font color="Olive" title="b6">i</font>, <font color="Olive" title="b7">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">j</font> &amp; <font color="Olive" title="b4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b5">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">N1</font>,<font color="Olive" title="b3">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b5">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:180_2_2_2"/><span class="lab"><font color="Green" title="E162">A42</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p9</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> st <br/> for <font color="Olive" title="b2">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b3">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b4">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H1</font><br/>  for <font color="Olive" title="b5">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">H2</font><br/>  for <font color="Olive" title="b6">i</font>, <font color="Olive" title="b7">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">j</font> &amp; <font color="Olive" title="b4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b5">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">N1</font>,<font color="Olive" title="b3">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b5">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><a NAME="E2:180_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E163">A43</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="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_2_2"><span class="lab"><font color="Green" title="E159">A39</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">p9</font> = <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  as   <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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> ;<br/><span class="kw">take </span><font color="Maroon" title="c11">p9</font> = <font color="Maroon" title="c10">p9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font><br/>  for <font color="Olive" title="b2">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b3">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H1</font><br/>  for <font color="Olive" title="b4">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H2</font><br/>  for <font color="Olive" title="b5">i</font>, <font color="Olive" title="b6">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">j</font> &amp; <font color="Olive" title="b3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N1</font>,<font color="Olive" title="b2">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b4">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c12">H1</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font><br/>  for <font color="Olive" title="b2">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c12">H1</font><br/>  for <font color="Olive" title="b3">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H2</font><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b1">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b2">N1</font>,<font color="Olive" title="b1">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c13">H2</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N1</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c12">H1</font><br/>  for <font color="Olive" title="b2">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c13">H2</font><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b2">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c14">N1</font> be   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c12">H1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N2</font> being   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c13">H2</font><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c15">N2</font> be   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c13">H2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c16">i</font>, <font color="Maroon" title="c17">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">assume </span><a NAME="E4:180_2_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><a NAME="E5:180_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">A44</font></span>: 
<font color="Maroon" title="c16">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="c4">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_2_2_2"><span class="lab"><font color="Green" title="E163">A43</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">assume </span><a NAME="E6:180_2_2_2"/><span class="lab"><font color="Green" title="E165">A45</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> &amp; <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:180_2_2_2"/><span class="lab"><font color="Green" title="E166">A46</font></span>: 
<font color="Maroon" title="c12">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">and </span><br/><a NAME="E8:180_2_2_2"/><span class="lab"><font color="Green" title="E167">A47</font></span>: 
<font color="Maroon" title="c13">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:180_2_2_2"/><span class="lab"><font color="Green" title="E168">A48</font></span>: 
<font color="Maroon" title="c14">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:180_2_2_2"/><span class="lab"><font color="Green" title="E169">A49</font></span>: 
<font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><a NAME="E11:180_2_2_2"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:180_2_2_2"><span class="lab"><font color="Green" title="E164">A44</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E12:180_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:180_2_2_2"><span class="lab"><font color="Green" title="E165">A45</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E13:180_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">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> <font color="Maroon" title="c9">n</font>
 ;<br/><a NAME="E14:180_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">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="c5">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:180"><span class="lab"><font color="Green" title="E157">A30</font></span></a>, <a class="txt" href="#E2:180_2_2"><span class="lab"><font color="Green" title="E159">A39</font></span></a>, <a class="txt" href="#E1:180_2_2_2"><span class="lab"><font color="Green" title="E162">A42</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="E15:180_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A50</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="c5">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:180_2_2"><span class="lab"><font color="Green" title="E160">A40</font></span></a>, <a class="txt" href="#E8:180_2_2_2"><span class="lab"><font color="Green" title="E167">A47</font></span></a>, <a class="txt" href="#E10:180_2_2_2"><span class="lab"><font color="Green" title="E169">A49</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><a NAME="E16:180_2_2_2"/>
<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">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_2_2"><span class="lab"><font color="Green" title="E158">A38</font></span></a>, <a class="txt" href="#E5:180_2_2_2"><span class="lab"><font color="Green" title="E164">A44</font></span></a>, <a class="txt" href="#E7:180_2_2_2"><span class="lab"><font color="Green" title="E166">A46</font></span></a>, <a class="txt" href="#E9:180_2_2_2"><span class="lab"><font color="Green" title="E168">A48</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><span class="kw">hence </span><a NAME="E17:180_2_2_2"/>
<font color="Maroon" title="c12">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">N1</font>,<font color="Maroon" title="c13">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c15">N2</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="#E5:180"><span class="lab"><font color="Green" title="E156">A29</font></span></a>, <a class="txt" href="#E5:180_2_2_2"><span class="lab"><font color="Green" title="E164">A44</font></span></a>, <a class="txt" href="#E6:180_2_2_2"><span class="lab"><font color="Green" title="E165">A45</font></span></a>, <a class="txt" href="#E15:180_2_2_2"><span class="lab"><font color="Green" title="E170">A50</font></span></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:180_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:180"><span class="lab"><font color="Green" title="E157">A30</font></span></a>, <a class="txt" href="#E2:180_2_2"><span class="lab"><font color="Green" title="E159">A39</font></span></a>, <a class="txt" href="#E3:180_2_2"><span class="lab"><font color="Green" title="E160">A40</font></span></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def36</a></span>;<br/><span class="kw">hence </span><a NAME="E6:180_2_2"/>
<font color="Maroon" title="c4">s1</font> <a href="group_9.html#R5" title="GROUP_9:pred.5">is_equivalent_with</a> <font color="Maroon" title="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:180_2_2"><span class="lab"><font color="Green" title="E161">A41</font></span></a>, <a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
