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<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G1</font>, <font color="Maroon" title="c3">G2</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c4">s1</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G1</font>;<br/><b>let </b><font color="Maroon" title="c5">s2</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c3">G2</font>;<br/>







<b>assume </b><a NAME="E1:184"/><i><font color="Green" title="E157">A1</font></i>: 
<font color="Maroon" title="c4">s1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E2:184"/><i><font color="Green" title="E158">A2</font></i>: 
<font color="Maroon" title="c5">s2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<a NAME="E3:184"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font> st <br/>for <font color="Olive" title="b3">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="b4">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="b5">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="b3">H1</font><br/> for <font color="Olive" title="b6">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="b4">H2</font><br/> for <font color="Olive" title="b7">i</font>, <font color="Olive" title="b8">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="b7">i</font> &amp; <font color="Olive" title="b7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i</font> &amp; <font color="Olive" title="b3">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="b7">i</font> &amp; <font color="Olive" title="b4">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="b8">j</font> &amp; <font color="Olive" title="b5">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="b7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">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="b8">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b3">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b5">N1</font>,<font color="Olive" title="b4">H2</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b6">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="group_9.html#E1:184"><i><font color="Green" title="E157">A1</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E4:184"/>
<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>
 <b>by </b><i><a class="txt" href="group_9.html#E1:184"><i><font color="Green" title="E157">A1</font></i></a>, <a class="txt" href="group_9.html#E2:184"><i><font color="Green" title="E158">A2</font></i></a>, <a class="ref" href="group_9.html#D35" target="_self" title="GROUP_9:def.35">Def35</a></i>;<br/>


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