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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b2">s1</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b3">fs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b4">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="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b2">s1</font>,<font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b3">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a></span><br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s1</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">fs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b3">i</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="Olive" title="b1">s1</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">s1</font> &amp; <font color="Olive" title="b1">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">s1</font>,<font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b2">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a></span><br/><b>let </b><font color="Maroon" title="c3">s1</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">fs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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="c3">s1</font> &amp; <font color="Olive" title="b2">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="c3">s1</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b2">i</font> holds <br/><font color="Olive" title="b1">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a></span><br/><b>let </b><font color="Maroon" title="c4">fs</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</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="c3">s1</font> &amp; <font color="Olive" title="b1">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="c3">s1</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a></span><br/><b>let </b><font color="Maroon" title="c5">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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="c3">s1</font> &amp; <font color="Maroon" title="c5">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="c3">s1</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font> implies <font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a> )</span><br/>









<b>assume </b><a NAME="E1:167"/><i><font color="Green" title="E140">A1</font></i>: 
<font color="Maroon" title="c5">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="c3">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">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="c3">s1</font> or  not <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or  not <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a> )</span><br/>

<b>assume </b><a NAME="E2:167"/><i><font color="Green" title="E141">A2</font></i>: 
<font color="Maroon" title="c5">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="c3">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or  not <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a> )</span><br/>

<b>assume </b><a NAME="E3:167"/><i><font color="Green" title="E142">A3</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a> )</span><br/>

<b>assume </b><a NAME="E4:167"/><i><font color="Green" title="E143">A4</font></i>: 
<font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a></span><br/>

<a NAME="E5:167"/><i><font color="Green" title="E144">A5</font></i>: 
<font color="Maroon" title="c5">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="c3">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:167"><i><font color="Green" title="E140">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:167"/><b>then </b><i><font color="Green" title="E145">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">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="c3">s1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:167"/><b>then </b><i><font color="Green" title="E146">A7</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:167"/>
<font color="Maroon" title="c5">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="c3">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E2:167"><i><font color="Green" title="E141">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:167"/><b>then </b><i><font color="Green" title="E147">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:167"/><i><font color="Green" title="E148">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:167"><i><font color="Green" title="E140">A1</font></i></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/>
<a NAME="E12:167"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">fs</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E7:167"><i><font color="Green" title="E146">A7</font></i></a>, <a class="txt" href="group_9.html#E9:167"><i><font color="Green" title="E147">A8</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:167"/><b>then </b><i><font color="Green" title="E149">A10</font></i>: 
1 <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">fs</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:167_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_9.html#E13:167"><i><font color="Green" title="E149">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1"><b>suppose </b></a><a NAME="E1:167_1_1"/><i><font color="Green" title="E150">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a></span><br/><div class="add"><a NAME="E2:167_1_1"/><b>then </b><i><font color="Green" title="E151">A12</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E5:167"><i><font color="Green" title="E144">A5</font></i></a>, <a class="txt" href="group_9.html#E9:167"><i><font color="Green" title="E147">A8</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:167_1_1"/><b>then </b><i><font color="Green" title="E152">A13</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:167_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:167_1_1"/><b>then </b><i><font color="Green" title="E153">A14</font></i>: 
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="c3">s1</font>
 ;<br/><a NAME="E6:167_1_1"/><i><font color="Green" title="E154">A15</font></i>: 
( <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/><a NAME="E7:167_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">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="c3">s1</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <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">fs</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="txt" href="group_9.html#E2:167_1_1"><i><font color="Green" title="E151">A12</font></i></a>, <a class="txt" href="group_9.html#E5:167_1_1"><i><font color="Green" title="E153">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>then </b><i><font color="Green" title="E155">A16</font></i>: <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font></span>)</span> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>
<br/>.= 
 <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
<b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">n</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">fs</font> &amp; <font color="Maroon" title="c7">n</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">fs</font> implies for <font color="Olive" title="b1">H1</font>, <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="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H2</font> is  <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> )</span><br/><b>assume </b><a NAME="E1:167_1_1_2"/>
<font color="Maroon" title="c7">n</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">fs</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">n</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">fs</font> implies for <font color="Olive" title="b1">H1</font>, <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="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H2</font> is  <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> )</span><br/><a NAME="E2:167_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">n</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> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:167_1_1_2"/><b>then </b><i><font color="Green" title="E156">A17</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>assume </b><a NAME="E4:167_1_1_2"/>
<font color="Maroon" title="c7">n</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">fs</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H1</font>, <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="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H2</font> is  <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></span><br/><a NAME="E5:167_1_1_2"/><b>then </b><i><font color="Green" title="E157">A18</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="txt" href="group_9.html#E3:167_1_1_2"><i><font color="Green" title="E156">A17</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>let </b><font color="Maroon" title="c8">H1</font>, <font color="Maroon" title="c9">H2</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c9">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c9">H2</font> is  <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="c8">H1</font> )</span><br/><b>assume </b><a NAME="E6:167_1_1_2"/>
( <font color="Maroon" title="c8">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c9">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H2</font> is  <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="c8">H1</font></span><br/><b>thus </b><a NAME="E7:167_1_1_2"/>
<font color="Maroon" title="c9">H2</font> is  <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="c8">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E5:167_1_1_2"><i><font color="Green" title="E157">A18</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E10:167_1_1"/>
<font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a>
 <b>by </b><i><a class="txt" href="group_9.html#E3:167"><i><font color="Green" title="E142">A3</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_1"><i><font color="Green" title="E150">A11</font></i></a>, <a class="txt" href="group_9.html#E3:167_1_1"><i><font color="Green" title="E152">A13</font></i></a>, <a class="txt" href="group_9.html#E6:167_1_1"><i><font color="Green" title="E154">A15</font></i></a>, <a class="txt" href="group_9.html#E8:167_1_1"><i><font color="Green" title="E155">A16</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2"><b>suppose </b></a><a NAME="E1:167_1_2"/><i><font color="Green" title="E150">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a></span><br/><div class="add"><a NAME="E2:167_1_2"/><i><font color="Green" title="E151">A20</font></i>: 
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:167_1_2_1_1"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_9.html#E6:167"><i><font color="Green" title="E145">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:167_1_2_1_1_1"/><i><font color="Green" title="E152">A21</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span><br/><div class="add"><a NAME="E2:167_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:167"><i><font color="Green" title="E140">A1</font></i></a>, <a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2"><i><font color="Green" title="E150">A19</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><b>hence </b><a NAME="E3:167_1_2_1_1_1"/>
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:167"><i><font color="Green" title="E142">A3</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2_1_1_1"><i><font color="Green" title="E152">A21</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:167_1_2_1_1_2"/><i><font color="Green" title="E152">A22</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c5">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2_1_1_2"><i><font color="Green" title="E152">A22</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></i>
;<br/><b>hence </b><a NAME="E4:167_1_2_1_1_2"/>
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:167_1_2"/>
<font color="Maroon" title="c5">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">fs</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E9:167"><i><font color="Green" title="E147">A8</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:167_1_2"/><b>then </b>
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:167"><i><font color="Green" title="E140">A1</font></i></a>, <a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><a NAME="E5:167_1_2"/><b>then </b><i><font color="Green" title="E152">A23</font></i>: 
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">n</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">fs</font> &amp; <font color="Maroon" title="c7">n</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">fs</font> implies for <font color="Olive" title="b1">H1</font>, <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="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive">b<sub>5</sub></font> is  <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">b<sub>4</sub></font> )</span><br/><b>assume </b><a NAME="E1:167_1_2_2"/>
( <font color="Maroon" title="c7">n</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">fs</font> &amp; <font color="Maroon" title="c7">n</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">fs</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H1</font>, <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="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive">b<sub>5</sub></font> is  <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">b<sub>4</sub></font></span><br/><a NAME="E2:167_1_2_2"/><b>then </b><i><font color="Green" title="E153">A24</font></i>: 
( <font color="Maroon" title="c7">n</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">fs</font></span>)</span> &amp; <font color="Maroon" title="c7">n</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">fs</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:167_1_2_2"/><i><font color="Green" title="E154">A25</font></i>: 
<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="c4">fs</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="c4">fs</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:167_1_2_2"/><b>then </b><i><font color="Green" title="E155">A26</font></i>: 
 <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">fs</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><b>let </b><font color="Maroon" title="c8">H1</font>, <font color="Maroon" title="c9">H2</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c9">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Olive">b<sub>3</sub></font> is  <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">b<sub>2</sub></font> )</span><br/><b>assume </b><a NAME="E5:167_1_2_2"/><i><font color="Green" title="E156">A27</font></i>: 
<font color="Maroon" title="c8">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Olive">b<sub>3</sub></font> is  <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">b<sub>2</sub></font> )</span><br/><b>assume </b><a NAME="E6:167_1_2_2"/><i><font color="Green" title="E157">A28</font></i>: 
<font color="Maroon" title="c9">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>3</sub></font> is  <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">b<sub>2</sub></font></span><br/><b>reconsider </b><font color="Maroon" title="c10">n1</font> = <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E8:167_1_2_2"/>
<font color="Maroon" title="c10">n1</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">fs</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:167_1_2_2"><i><font color="Green" title="E153">A24</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:167_1_2_2"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E3:167_1_2_2"><i><font color="Green" title="E154">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:167_1_2_2"/><b>then </b><i><font color="Green" title="E158">A29</font></i>: 
<font color="Maroon" title="c10">n1</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> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:167_1_2_2"/><i><font color="Green" title="E159">A30</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E2:167_1_2_2"><i><font color="Green" title="E153">A24</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:167_1_2_2"/>
<font color="Maroon" title="c7">n</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="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:167_1_2_2"><i><font color="Green" title="E153">A24</font></i></a>, <a class="txt" href="group_9.html#E4:167_1_2_2"><i><font color="Green" title="E155">A26</font></i></a></i>;<br/><a NAME="E13:167_1_2_2"/><b>then </b><i><font color="Green" title="E160">A31</font></i>: 
<font color="Maroon" title="c7">n</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="c3">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:167_1_2_2"/>
<font color="Maroon" title="c10">n1</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="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:167_1_2_2"><i><font color="Green" title="E153">A24</font></i></a>, <a class="txt" href="group_9.html#E4:167_1_2_2"><i><font color="Green" title="E155">A26</font></i></a></i>;<br/><a NAME="E15:167_1_2_2"/><b>then </b><i><font color="Green" title="E161">A32</font></i>: 
<font color="Maroon" title="c10">n1</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="c3">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:167_1_2_2"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E10:167_1_2_2"><i><font color="Green" title="E158">A29</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><a NAME="E17:167_1_2_2"/><b>then </b><i><font color="Green" title="E162">A33</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:167_1_2_2"/><i><font color="Green" title="E163">A34</font></i>: 
<font color="Maroon" title="c10">n1</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"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/><a NAME="E19:167_1_2_2"/><i><font color="Green" title="E164">A35</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E2:167_1_2_2"><i><font color="Green" title="E153">A24</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c7">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:167_1_2_2_1"/>
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> or <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:167_1_2_2_1_1"/><i><font color="Green" title="E165">A36</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>3</sub></font> is  <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">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:167_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E166">A37</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">n'</font> = <font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E4:167_1_2_2_1_1"/><i><font color="Green" title="E167">A38</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2_2_1_1"><i><font color="Green" title="E165">A36</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:167_1_2_2_1_1_1"/>
( <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> or <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font> )
 </span><b>by </b><i><a class="txt" href="group_9.html#E2:167_1_2_2_1_1"><i><font color="Green" title="E166">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:167_1_2_2_1_1_1_1"/><i><font color="Green" title="E168">A39</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>3</sub></font> is  <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">b<sub>2</sub></font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c15">n1'</font> = <font color="Maroon" title="c10">n1</font>, <font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:167_1_2_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2_2_1_1_1_1"><i><font color="Green" title="E168">A39</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></i>;<br/><b>hence </b><a NAME="E4:167_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c9">H2</font> is  <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="c8">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E5:167_1_2_2"><i><font color="Green" title="E156">A27</font></i></a>, <a class="txt" href="group_9.html#E6:167_1_2_2"><i><font color="Green" title="E157">A28</font></i></a>, <a class="txt" href="group_9.html#E13:167_1_2_2"><i><font color="Green" title="E160">A31</font></i></a>, <a class="txt" href="group_9.html#E15:167_1_2_2"><i><font color="Green" title="E161">A32</font></i></a>, <a class="txt" href="group_9.html#E4:167_1_2_2_1_1"><i><font color="Green" title="E167">A38</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:167_1_2_2_1_1_1_2"/><i><font color="Green" title="E168">A40</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>3</sub></font> is  <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">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:167_1_2_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:167"><i><font color="Green" title="E140">A1</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E11:167_1_2_2"><i><font color="Green" title="E159">A30</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><b>hence </b><a NAME="E3:167_1_2_2_1_1_1_2"/>
<font color="Maroon" title="c9">H2</font> is  <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="c8">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:167"><i><font color="Green" title="E142">A3</font></i></a>, <a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E5:167_1_2_2"><i><font color="Green" title="E156">A27</font></i></a>, <a class="txt" href="group_9.html#E6:167_1_2_2"><i><font color="Green" title="E157">A28</font></i></a>, <a class="txt" href="group_9.html#E13:167_1_2_2"><i><font color="Green" title="E160">A31</font></i></a>, <a class="txt" href="group_9.html#E15:167_1_2_2"><i><font color="Green" title="E161">A32</font></i></a>, <a class="txt" href="group_9.html#E4:167_1_2_2_1_1"><i><font color="Green" title="E167">A38</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2_2_1_1_1_2"><i><font color="Green" title="E168">A40</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:167_1_2_2_1_2"/><i><font color="Green" title="E165">A41</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>3</sub></font> is  <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">b<sub>2</sub></font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c13">n'</font> = <font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E3:167_1_2_2_1_2"/><i><font color="Green" title="E166">A42</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:167"><i><font color="Green" title="E140">A1</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E19:167_1_2_2"><i><font color="Green" title="E164">A35</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2_2_1_2"><i><font color="Green" title="E165">A41</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">n1'</font> = <font color="Maroon" title="c10">n1</font>, <font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c6">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E5:167_1_2_2_1_2"/>
<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="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:167_1_2_2_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <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> <font color="Maroon" title="c3">s1</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="c15">n1'</font> &amp; <font color="Maroon" title="c15">n1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:167"><i><font color="Green" title="E140">A1</font></i></a>, <a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E11:167"><i><font color="Green" title="E148">A9</font></i></a>, <a class="txt" href="group_9.html#E2:167_1_2_2"><i><font color="Green" title="E153">A24</font></i></a>, <a class="txt" href="group_9.html#E1:167_1_2_2_1_2"><i><font color="Green" title="E165">A41</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:167_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n1'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><b>hence </b><a NAME="E8:167_1_2_2_1_2"/>
<font color="Maroon" title="c9">H2</font> is  <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="c8">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:167"><i><font color="Green" title="E143">A4</font></i></a>, <a class="txt" href="group_9.html#E5:167_1_2_2"><i><font color="Green" title="E156">A27</font></i></a>, <a class="txt" href="group_9.html#E6:167_1_2_2"><i><font color="Green" title="E157">A28</font></i></a>, <a class="txt" href="group_9.html#E15:167_1_2_2"><i><font color="Green" title="E161">A32</font></i></a>, <a class="txt" href="group_9.html#E17:167_1_2_2"><i><font color="Green" title="E162">A33</font></i></a>, <a class="txt" href="group_9.html#E18:167_1_2_2"><i><font color="Green" title="E163">A34</font></i></a>, <a class="txt" href="group_9.html#E3:167_1_2_2_1_2"><i><font color="Green" title="E166">A42</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:167_1_2"/>
<font color="Maroon" title="c4">fs</font> is <a href="group_9.html#V11" title="GROUP_9:attr.11">composition_series</a>
 <b>by </b><i><a class="txt" href="group_9.html#E2:167_1_2"><i><font color="Green" title="E151">A20</font></i></a>, <a class="txt" href="group_9.html#E5:167_1_2"><i><font color="Green" title="E152">A23</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
