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





<b>assume </b><a NAME="E1:192"/><i><font color="Green" title="E165">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<div><i><font color="Green" title="E166">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="192_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:192_1"/><i><font color="Green" title="E167">A3</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a>
 ;<br/><b>assume </b><a NAME="E2:192_1"/>
 ex <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <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> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  not  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c4">i</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:192_1"/><i><font color="Green" title="E168">A4</font></i>: 
<font color="Maroon" title="c4">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="c3">s1</font></span>)</span>
 <b>and </b><br/><a NAME="E5:192_1"/><i><font color="Green" title="E169">A5</font></i>: 
<span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is  not  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>
 ;<br/><a NAME="E6:192_1"/><i><font color="Green" title="E170">A6</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> <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="c3">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:192"><i><font color="Green" title="E165">A1</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E7:192_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c3">s1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c3">s1</font></span>)</span></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="E8:192_1"/><b>then </b><i><font color="Green" title="E171">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <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#E1:192"><i><font color="Green" title="E165">A1</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E9:192_1"/><i><font color="Green" title="E172">A8</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E4:192_1"><i><font color="Green" title="E168">A4</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="E10:192_1"/><b>then </b><i><font color="Green" title="E173">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> )
 <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="E11:192_1"/><b>then </b><i><font color="Green" title="E174">A10</font></i>: 
<font color="Maroon" title="c4">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="txt" href="group_9.html#E8:192_1"><i><font color="Green" title="E171">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:192_1"/><b>then </b>
<font color="Maroon" title="c4">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#E10:192_1"><i><font color="Green" title="E173">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:192_1"/><b>then </b><i><font color="Green" title="E175">A11</font></i>: 
<font color="Maroon" title="c4">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>
 <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="E14:192_1"/><i><font color="Green" title="E176">A12</font></i>: 
0 <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="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="E15:192_1"/><i><font color="Green" title="E177">A13</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="txt" href="group_9.html#E6:192_1"><i><font color="Green" title="E170">A6</font></i></a>, <a class="txt" href="group_9.html#E10:192_1"><i><font color="Green" title="E173">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="E16:192_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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#E14:192_1"><i><font color="Green" title="E176">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:192_1"/><b>then </b><i><font color="Green" title="E178">A14</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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/><b>then reconsider </b><font color="Maroon" title="c5">H1</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c6">N1</font> = <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="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">G</font> <b>by </b><i><a class="txt" href="group_9.html#E13:192_1"><i><font color="Green" title="E175">A11</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">H1</font> = <font color="Maroon" title="c5">H1</font>, <font color="Maroon" title="c8">N1</font> = <font color="Maroon" title="c6">N1</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">N1</font> = <font color="Maroon" title="c8">N1</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c7">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E13:192_1"><i><font color="Green" title="E175">A11</font></i></a>, <a class="txt" href="group_9.html#E17:192_1"><i><font color="Green" title="E178">A14</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/><a NAME="E21:192_1"/><i><font color="Green" title="E179">A15</font></i>: 
<font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font> is  not  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>
 <b>by </b><i><a class="txt" href="group_9.html#E5:192_1"><i><font color="Green" title="E169">A5</font></i></a>, <a class="txt" href="group_9.html#E1:192"><i><font color="Green" title="E165">A1</font></i></a>, <a class="txt" href="group_9.html#E4:192_1"><i><font color="Green" title="E168">A4</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:192_1_1"/>
( <font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> or  ex <font color="Olive" title="b1">H</font> being <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font></span>)</span> &amp; <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="group_9.html#E21:192_1"><i><font color="Green" title="E179">A15</font></i></a>, <a class="ref" href="group_9.html#D13" target="_self" title="GROUP_9:def.13">Def13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_1"><b>suppose </b></a><a NAME="E1:192_1_1_1"/><i><font color="Green" title="E180">A16</font></i>: 
<font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/><div class="add"><a NAME="E2:192_1_1_1"/>
<font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1"><i><font color="Green" title="E167">A3</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>;<br/><b>hence </b><a NAME="E3:192_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_1"><i><font color="Green" title="E180">A16</font></i></a>, <a class="txt" href="group_9.html#E13:192_1"><i><font color="Green" title="E175">A11</font></i></a>, <a class="txt" href="group_9.html#E17:192_1"><i><font color="Green" title="E178">A14</font></i></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2"><b>suppose </b></a><a NAME="E1:192_1_1_2"/>
 ex <font color="Olive" title="b1">H</font> being <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font></span>)</span> &amp; <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font></span>)</span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">H</font> being  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font><b> such that </b><br/><a NAME="E3:192_1_1_2"/><i><font color="Green" title="E180">A17</font></i>: 
( <font color="Maroon" title="c10">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font></span>)</span> &amp; <font color="Maroon" title="c10">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c9">N1</font></span>)</span> )
 ;<br/><a NAME="E4:192_1_1_2"/>
<font color="Maroon" title="c9">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <span class="p1">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c9">N1</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_9.html#T48" target="_self" title="GROUP_9:th.48">Th48</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">N2</font> being  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c7">H1</font><b> such that </b><br/><a NAME="E6:192_1_1_2"/><i><font color="Green" title="E181">A18</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c9">N1</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font> &amp; ( <font color="Maroon" title="c10">H</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> implies ( <font color="Maroon" title="c9">N1</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="c11">N2</font> &amp; <font color="Maroon" title="c11">N2</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> ) ) )
 <b>by </b><i><a class="ref" href="group_9.html#T78" target="_self" title="GROUP_9:th.78">Th78</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c4">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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E8:192_1_1_2"/>
<font color="Maroon" title="c11">N2</font> is  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="group_9.html#T11" target="_self" title="GROUP_9:th.11">Th11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">N2</font> = <font color="Maroon" title="c11">N2</font> 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">G</font> <b>by </b><i><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></i>;<br/><b>set </b><font color="Maroon" title="c14">s2</font> =  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font>;<br/><a NAME="E10:192_1_1_2"/><i><font color="Green" title="E182">A19</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; not <font color="Maroon" title="c3">s1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:192"><i><font color="Green" title="E165">A1</font></i></a>, <a class="txt" href="group_9.html#E9:192_1"><i><font color="Green" title="E172">A8</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:192_1_1_2"/>
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="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:192"><i><font color="Green" title="E165">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:192_1_1_2"/><b>then </b><i><font color="Green" title="E183">A20</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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/><a NAME="E13:192_1_1_2"/><b>then </b>
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> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E14:192_1_1_2"/><b>then </b><i><font color="Green" title="E184">A21</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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</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="E15:192_1_1_2"/>
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#E1:192"><i><font color="Green" title="E165">A1</font></i></a></i>;<br/><a NAME="E16:192_1_1_2"/><b>then </b><i><font color="Green" title="E185">A22</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>
 <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="E17:192_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <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="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E12:192_1_1_2"><i><font color="Green" title="E183">A20</font></i></a></i>;<br/><a NAME="E18:192_1_1_2"/><b>then </b><i><font color="Green" title="E186">A23</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <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="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</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="E19:192_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</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#E1:192"><i><font color="Green" title="E165">A1</font></i></a></i>;<br/><a NAME="E20:192_1_1_2"/><b>then </b><i><font color="Green" title="E187">A24</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#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/><i><font color="Green" title="E188">A25</font></i>: <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="group_9.html#E14:192_1_1_2"><i><font color="Green" title="E184">A21</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="group_9.html#E10:192_1_1_2"><i><font color="Green" title="E182">A19</font></i></a>, <a class="ref" href="finseq_5.html#T78" title="FINSEQ_5:th.78">FINSEQ_5:78</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#E16:192_1_1_2"><i><font color="Green" title="E185">A22</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <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>
;<br/><i><font color="Green" title="E189">A26</font></i>: <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <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> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E18:192_1_1_2"><i><font color="Green" title="E186">A23</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#E15:192_1"><i><font color="Green" title="E177">A13</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<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#E20:192_1_1_2"><i><font color="Green" title="E187">A24</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</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="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>
;<br/><a NAME="E23:192_1_1_2"/><i><font color="Green" title="E190">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/><a NAME="E24:192_1_1_2"/><b>then </b><i><font color="Green" title="E191">A28</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="192_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">j</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:192_1_1_2_3"/><i><font color="Green" title="E192">A29</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span>
 ;<br/><a NAME="E2:192_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c15">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="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:192_1_1_2_3"/><b>then </b><i><font color="Green" title="E193">A30</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>assume </b><a NAME="E4:192_1_1_2_3"/><i><font color="Green" title="E194">A31</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span>
 ;<br/><a NAME="E5:192_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:192_1_1_2_3"/><b>then </b><i><font color="Green" title="E195">A32</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>let </b><font color="Maroon" title="c16">H1'</font>, <font color="Maroon" title="c17">H2'</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>;<br/><b>assume </b><a NAME="E7:192_1_1_2_3"/><i><font color="Green" title="E196">A33</font></i>: 
<font color="Maroon" title="c16">H1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
 ;<br/><b>assume </b><a NAME="E8:192_1_1_2_3"/><i><font color="Green" title="E197">A34</font></i>: 
<font color="Maroon" title="c17">H2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E9:192_1_1_2_3"/>
( <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">i</font> )
 <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="E10:192_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:192_1_1_2_3"/><b>then </b><i><font color="Green" title="E198">A35</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">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#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:192_1_1_2_3_1"/>
( <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> )
 </span><b>by </b><i><a class="txt" href="group_9.html#E11:192_1_1_2_3"><i><font color="Green" title="E198">A35</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="192_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:192_1_1_2_3_1_1"/><i><font color="Green" title="E199">A36</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><div class="add"><a NAME="E2:192_1_1_2_3_1_1"/>
<font color="Maroon" title="c12">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="txt" href="group_9.html#E10:192_1"><i><font color="Green" title="E173">A9</font></i></a>, <a class="txt" href="group_9.html#E8:192_1"><i><font color="Green" title="E171">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:192_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E200">A37</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></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="c12">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E4:192_1_1_2_3_1_1"/><i><font color="Green" title="E201">A38</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></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="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E5:192_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font>
 <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="E6:192_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_3_1_1"><i><font color="Green" title="E199">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:192_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c15">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"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2_3_1_1"><i><font color="Green" title="E200">A37</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="group_9.html#E3:192_1_1_2_3"><i><font color="Green" title="E193">A30</font></i></a></i>;<br/><a NAME="E8:192_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E202">A39</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</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="E9:192_1_1_2_3_1_1"/>
<font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_3_1_1"><i><font color="Green" title="E199">A36</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:192_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2_3_1_1"><i><font color="Green" title="E200">A37</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="group_9.html#E6:192_1_1_2_3"><i><font color="Green" title="E195">A32</font></i></a></i>;<br/><a NAME="E11:192_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E203">A40</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</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/><i><font color="Green" title="E204">A41</font></i>: <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_3"><i><font color="Green" title="E192">A29</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E8:192_1_1_2_3_1_1"><i><font color="Green" title="E202">A39</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E8:192_1_1_2_3_1_1"><i><font color="Green" title="E202">A39</font></i></a>, <a class="txt" href="group_9.html#E4:192_1_1_2_3_1_1"><i><font color="Green" title="E201">A38</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E4:192_1_1_2_3"><i><font color="Green" title="E194">A31</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E11:192_1_1_2_3_1_1"><i><font color="Green" title="E203">A40</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<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">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E11:192_1_1_2_3_1_1"><i><font color="Green" title="E203">A40</font></i></a>, <a class="txt" href="group_9.html#E4:192_1_1_2_3_1_1"><i><font color="Green" title="E201">A38</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E14:192_1_1_2_3_1_1"/>
<font color="Maroon" title="c17">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="c16">H1'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E12:192_1_1_2_3_1_1"><i><font color="Green" title="E204">A41</font></i></a>, <a class="txt" href="group_9.html#E7:192_1_1_2_3"><i><font color="Green" title="E196">A33</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_3"><i><font color="Green" title="E197">A34</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_3_1_1"><i><font color="Green" title="E202">A39</font></i></a>, <a class="txt" href="group_9.html#E4:192_1_1_2_3_1_1"><i><font color="Green" title="E201">A38</font></i></a>, <a class="txt" href="group_9.html#E11:192_1_1_2_3_1_1"><i><font color="Green" title="E203">A40</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:192_1_1_2_3_1_2"/><i><font color="Green" title="E199">A42</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E200">A43</font></i>: <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</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#E4:192_1_1_2_3"><i><font color="Green" title="E194">A31</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c13">N2</font>
<b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="finseq_5.html#T76" title="FINSEQ_5:th.76">FINSEQ_5:76</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c18">N2</font> = <font color="Maroon" title="c13">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="c7">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2"><i><font color="Green" title="E181">A18</font></i></a></i>;<br/><a NAME="E4:192_1_1_2_3_1_2"/><i><font color="Green" title="E201">A44</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></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="c12">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a>, <a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a></i>;<br/><a NAME="E5:192_1_1_2_3_1_2"/><i><font color="Green" title="E202">A45</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></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="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E6:192_1_1_2_3_1_2"/>
<font color="Maroon" title="c15">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="c12">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_3_1_2"><i><font color="Green" title="E199">A42</font></i></a>, <a class="txt" href="group_9.html#E3:192_1_1_2_3"><i><font color="Green" title="E193">A30</font></i></a></i>;<br/><a NAME="E7:192_1_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E203">A46</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E4:192_1_1_2_3_1_2"><i><font color="Green" title="E201">A44</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_3"><i><font color="Green" title="E192">A29</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E7:192_1_1_2_3_1_2"><i><font color="Green" title="E203">A46</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E7:192_1_1_2_3_1_2"><i><font color="Green" title="E203">A46</font></i></a>, <a class="txt" href="group_9.html#E5:192_1_1_2_3_1_2"><i><font color="Green" title="E202">A45</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E9:192_1_1_2_3_1_2"/>
<font color="Maroon" title="c17">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="c16">H1'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:192_1_1_2_3_1_2"><i><font color="Green" title="E200">A43</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_3"><i><font color="Green" title="E197">A34</font></i></a>, <a class="txt" href="group_9.html#E6:192_1_1_2"><i><font color="Green" title="E181">A18</font></i></a>, <a class="txt" href="group_9.html#E1:192_1_1_2_3_1_2"><i><font color="Green" title="E199">A42</font></i></a>, <a class="txt" href="group_9.html#E7:192_1_1_2_3"><i><font color="Green" title="E196">A33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_3_1_3"><b>suppose </b></a><a NAME="E1:192_1_1_2_3_1_3"/><i><font color="Green" title="E199">A47</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:192_1_1_2_3_1_3"/>
<font color="Maroon" title="c12">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="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:192_1_1_2_3_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E14:192_1"><i><font color="Green" title="E176">A12</font></i></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/><a NAME="E4:192_1_1_2_3_1_3"/><b>then </b>
<font color="Maroon" title="c12">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> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:192_1_1_2_3_1_3"/><b>then </b>
<font color="Maroon" title="c12">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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</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/><b>then </b><i><font color="Green" title="E200">A48</font></i>: <font color="Maroon" title="c16">H1'</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</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#E7:192_1_1_2_3"><i><font color="Green" title="E196">A33</font></i></a>, <a class="txt" href="group_9.html#E1:192_1_1_2_3_1_3"><i><font color="Green" title="E199">A47</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c13">N2</font>
<b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="finseq_5.html#T76" title="FINSEQ_5:th.76">FINSEQ_5:76</a></i>
;<br/><a NAME="E7:192_1_1_2_3_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E15:192_1"><i><font color="Green" title="E177">A13</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:192_1_1_2_3_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="txt" href="group_9.html#E14:192_1"><i><font color="Green" title="E176">A12</font></i></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/><a NAME="E9:192_1_1_2_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E10:192_1_1_2_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</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/><b>then </b><font color="Maroon" title="c17">H2'</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a>, <a class="txt" href="group_9.html#E8:192_1_1_2_3"><i><font color="Green" title="E197">A34</font></i></a>, <a class="txt" href="group_9.html#E1:192_1_1_2_3_1_3"><i><font color="Green" title="E199">A47</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</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#E15:192_1"><i><font color="Green" title="E177">A13</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon" title="c9">N1</font>
<b>by </b><i><a class="txt" href="group_9.html#E17:192_1"><i><font color="Green" title="E178">A14</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E12:192_1_1_2_3_1_3"/>
<font color="Maroon" title="c17">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="c16">H1'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2"><i><font color="Green" title="E181">A18</font></i></a>, <a class="txt" href="group_9.html#E6:192_1_1_2_3_1_3"><i><font color="Green" title="E200">A48</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_3_1_4"><b>suppose </b></a><a NAME="E1:192_1_1_2_3_1_4"/><i><font color="Green" title="E199">A49</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 ;<br/><div class="add"><a NAME="E2:192_1_1_2_3_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c15">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:192_1_1_2_3_1_4"/><b>then </b><i><font color="Green" title="E200">A50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>set </b><font color="Maroon" title="c18">j'</font> = <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><a NAME="E4:192_1_1_2_3_1_4"/>
0 <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="c12">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="E5:192_1_1_2_3_1_4"/><b>then </b><i><font color="Green" title="E201">A51</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_3_1_4"><i><font color="Green" title="E199">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:192_1_1_2_3_1_4"/><b>then </b><i><font color="Green" title="E202">A52</font></i>: 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">j'</font> = <font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="E8:192_1_1_2_3_1_4"/><i><font color="Green" title="E203">A53</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2_3"><i><font color="Green" title="E193">A30</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:192_1_1_2_3_1_4"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">j'</font>
 <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="E10:192_1_1_2_3_1_4"/><b>then </b><i><font color="Green" title="E204">A54</font></i>: 
<font color="Maroon" title="c12">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> <font color="Maroon" title="c19">j'</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#E3:192_1_1_2_3_1_4"><i><font color="Green" title="E200">A50</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:192_1_1_2_3_1_4"/><i><font color="Green" title="E205">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2_3"><i><font color="Green" title="E195">A32</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:192_1_1_2_3_1_4"/>
0 <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="c19">j'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2_3_1_4"><i><font color="Green" title="E202">A52</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:192_1_1_2_3_1_4"/><b>then </b>
<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="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E8:192_1_1_2_3_1_4"><i><font color="Green" title="E203">A53</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a></i>;<br/><a NAME="E14:192_1_1_2_3_1_4"/><b>then </b><i><font color="Green" title="E206">A56</font></i>: 
<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="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="E15:192_1_1_2_3_1_4"/>
<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="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E5:192_1_1_2_3_1_4"><i><font color="Green" title="E201">A51</font></i></a>, <a class="txt" href="group_9.html#E11:192_1_1_2_3_1_4"><i><font color="Green" title="E205">A55</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a></i>;<br/><a NAME="E16:192_1_1_2_3_1_4"/><b>then </b><i><font color="Green" title="E207">A57</font></i>: 
<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="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/><i><font color="Green" title="E208">A58</font></i>: <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
<b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_3"><i><font color="Green" title="E192">A29</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">j'</font>
<b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2_3_1_4"><i><font color="Green" title="E200">A50</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_3_1_4"><i><font color="Green" title="E203">A53</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j'</font>
<b>by </b><i><a class="txt" href="group_9.html#E14:192_1_1_2_3_1_4"><i><font color="Green" title="E206">A56</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <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#E4:192_1_1_2_3"><i><font color="Green" title="E194">A31</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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>
<b>by </b><i><a class="txt" href="group_9.html#E10:192_1_1_2_3_1_4"><i><font color="Green" title="E204">A54</font></i></a>, <a class="txt" href="group_9.html#E11:192_1_1_2_3_1_4"><i><font color="Green" title="E205">A55</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<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="c19">j'</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#E16:192_1_1_2_3_1_4"><i><font color="Green" title="E207">A57</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E19:192_1_1_2_3_1_4"/>
<font color="Maroon" title="c17">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="c16">H1'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E17:192_1_1_2_3_1_4"><i><font color="Green" title="E208">A58</font></i></a>, <a class="txt" href="group_9.html#E7:192_1_1_2_3"><i><font color="Green" title="E196">A33</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_3"><i><font color="Green" title="E197">A34</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a>, <a class="txt" href="group_9.html#E14:192_1_1_2_3_1_4"><i><font color="Green" title="E206">A56</font></i></a>, <a class="txt" href="group_9.html#E16:192_1_1_2_3_1_4"><i><font color="Green" title="E207">A57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">s2</font> =  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">N2</font> as   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E21:192_1_1_2"><i><font color="Green" title="E188">A25</font></i></a>, <a class="txt" href="group_9.html#E22:192_1_1_2"><i><font color="Green" title="E189">A26</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="192_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">j</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:192_1_1_2_4"/><i><font color="Green" title="E192">A59</font></i>: 
<font color="Maroon" title="c16">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="c15">s2</font>
 ;<br/><a NAME="E2:192_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c16">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="c15">s2</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="E3:192_1_1_2_4"/><b>then </b><i><font color="Green" title="E193">A60</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font> &amp; <font color="Maroon" title="c16">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="c15">s2</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>assume </b><a NAME="E4:192_1_1_2_4"/><i><font color="Green" title="E194">A61</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">s2</font>
 ;<br/><a NAME="E5:192_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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="c15">s2</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="E6:192_1_1_2_4"/><b>then </b><i><font color="Green" title="E195">A62</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c15">s2</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>let </b><font color="Maroon" title="c17">H1'</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c18">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="c17">H1'</font>;<br/><b>assume </b><a NAME="E7:192_1_1_2_4"/><i><font color="Green" title="E196">A63</font></i>: 
<font color="Maroon" title="c17">H1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
 ;<br/><b>assume </b><a NAME="E8:192_1_1_2_4"/><i><font color="Green" title="E197">A64</font></i>: 
<font color="Maroon" title="c18">N1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E9:192_1_1_2_4"/>
( <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">i</font> )
 <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="E10:192_1_1_2_4"/><b>then </b>
( <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:192_1_1_2_4"/><b>then </b><i><font color="Green" title="E198">A65</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">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#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:192_1_1_2_4"/><i><font color="Green" title="E199">A66</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1"><i><font color="Green" title="E167">A3</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:192_1_1_2_4_1"/>
( <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> )
 </span><b>by </b><i><a class="txt" href="group_9.html#E11:192_1_1_2_4"><i><font color="Green" title="E198">A65</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="192_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:192_1_1_2_4_1_1"/><i><font color="Green" title="E200">A67</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><div class="add"><a NAME="E2:192_1_1_2_4_1_1"/><i><font color="Green" title="E201">A68</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></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="c12">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:192_1_1_2_4_1_1"/><i><font color="Green" title="E202">A69</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></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="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E4:192_1_1_2_4_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font>
 <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="E5:192_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_4_1_1"><i><font color="Green" title="E200">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:192_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon" title="c16">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"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:192_1_1_2_4_1_1"><i><font color="Green" title="E201">A68</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="group_9.html#E3:192_1_1_2_4"><i><font color="Green" title="E193">A60</font></i></a></i>;<br/><a NAME="E7:192_1_1_2_4_1_1"/><b>then </b><i><font color="Green" title="E203">A70</font></i>: 
<font color="Maroon" title="c16">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"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</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="E8:192_1_1_2_4_1_1"/>
<font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_4_1_1"><i><font color="Green" title="E200">A67</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:192_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:192_1_1_2_4_1_1"><i><font color="Green" title="E201">A68</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="group_9.html#E6:192_1_1_2_4"><i><font color="Green" title="E195">A62</font></i></a></i>;<br/><a NAME="E10:192_1_1_2_4_1_1"/><b>then </b><i><font color="Green" title="E204">A71</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</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/><i><font color="Green" title="E205">A72</font></i>: <font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_4"><i><font color="Green" title="E192">A59</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E7:192_1_1_2_4_1_1"><i><font color="Green" title="E203">A70</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E7:192_1_1_2_4_1_1"><i><font color="Green" title="E203">A70</font></i></a>, <a class="txt" href="group_9.html#E3:192_1_1_2_4_1_1"><i><font color="Green" title="E202">A69</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E4:192_1_1_2_4"><i><font color="Green" title="E194">A61</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E10:192_1_1_2_4_1_1"><i><font color="Green" title="E204">A71</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<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="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E10:192_1_1_2_4_1_1"><i><font color="Green" title="E204">A71</font></i></a>, <a class="txt" href="group_9.html#E3:192_1_1_2_4_1_1"><i><font color="Green" title="E202">A69</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E13:192_1_1_2_4_1_1"/>
not <font color="Maroon" title="c17">H1'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c18">N1'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E11:192_1_1_2_4_1_1"><i><font color="Green" title="E205">A72</font></i></a>, <a class="txt" href="group_9.html#E7:192_1_1_2_4"><i><font color="Green" title="E196">A63</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_4"><i><font color="Green" title="E197">A64</font></i></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a>, <a class="txt" href="group_9.html#E7:192_1_1_2_4_1_1"><i><font color="Green" title="E203">A70</font></i></a>, <a class="txt" href="group_9.html#E10:192_1_1_2_4_1_1"><i><font color="Green" title="E204">A71</font></i></a>, <a class="txt" href="group_9.html#E3:192_1_1_2_4_1_1"><i><font color="Green" title="E202">A69</font></i></a>, <a class="txt" href="group_9.html#E12:192_1_1_2_4"><i><font color="Green" title="E199">A66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:192_1_1_2_4_1_2"/><i><font color="Green" title="E200">A73</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E201">A74</font></i>: <font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</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#E4:192_1_1_2_4"><i><font color="Green" title="E194">A61</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c13">N2</font>
<b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="finseq_5.html#T76" title="FINSEQ_5:th.76">FINSEQ_5:76</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c19">N2</font> = <font color="Maroon" title="c13">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="c7">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2"><i><font color="Green" title="E181">A18</font></i></a></i>;<br/><a NAME="E4:192_1_1_2_4_1_2"/><i><font color="Green" title="E202">A75</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span></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="c12">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E5:192_1_1_2_4_1_2"/><i><font color="Green" title="E203">A76</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></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="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E6:192_1_1_2_4_1_2"/>
<font color="Maroon" title="c16">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="c12">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_4_1_2"><i><font color="Green" title="E200">A73</font></i></a>, <a class="txt" href="group_9.html#E3:192_1_1_2_4"><i><font color="Green" title="E193">A60</font></i></a></i>;<br/><a NAME="E7:192_1_1_2_4_1_2"/><b>then </b><i><font color="Green" title="E204">A77</font></i>: 
<font color="Maroon" title="c16">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"><font color="Maroon" title="c3">s1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E4:192_1_1_2_4_1_2"><i><font color="Green" title="E202">A75</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green" title="E205">A78</font></i>: <font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_4"><i><font color="Green" title="E192">A59</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E7:192_1_1_2_4_1_2"><i><font color="Green" title="E204">A77</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
<b>by </b><i><a class="txt" href="group_9.html#E7:192_1_1_2_4_1_2"><i><font color="Green" title="E204">A77</font></i></a>, <a class="txt" href="group_9.html#E5:192_1_1_2_4_1_2"><i><font color="Green" title="E203">A76</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="192_1_1_2_4_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:192_1_1_2_4_1_2_3"/>
<font color="Maroon" title="c17">H1'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c18">N1'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/><a NAME="E2:192_1_1_2_4_1_2_3"/><b>then </b>
<font color="Maroon" title="c7">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">N2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E8:192_1_1_2_4_1_2"><i><font color="Green" title="E205">A78</font></i></a>, <a class="ref" href="group_9.html#T76" target="_self" title="GROUP_9:th.76">Th76</a>, <a class="txt" href="group_9.html#E1:192_1_1_2_4_1_2"><i><font color="Green" title="E200">A73</font></i></a>, <a class="txt" href="group_9.html#E7:192_1_1_2_4"><i><font color="Green" title="E196">A63</font></i></a>, <a class="txt" href="group_9.html#E2:192_1_1_2_4_1_2"><i><font color="Green" title="E201">A74</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_4"><i><font color="Green" title="E197">A64</font></i></a></i>;<br/><b>hence </b><a NAME="E3:192_1_1_2_4_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2"><i><font color="Green" title="E180">A17</font></i></a>, <a class="txt" href="group_9.html#E6:192_1_1_2"><i><font color="Green" title="E181">A18</font></i></a>, <a class="ref" href="group_9.html#T80" target="_self" title="GROUP_9:th.80">Th80</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:192_1_1_2_4_1_2"/>
not <font color="Maroon" title="c17">H1'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c18">N1'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_4_1_3"><b>suppose </b></a><a NAME="E1:192_1_1_2_4_1_3"/><i><font color="Green" title="E200">A79</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:192_1_1_2_4_1_3"/>
<font color="Maroon" title="c12">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="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:192_1_1_2_4_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">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="c15">s2</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E14:192_1"><i><font color="Green" title="E176">A12</font></i></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/><a NAME="E4:192_1_1_2_4_1_3"/><b>then </b>
<font color="Maroon" title="c12">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="c15">s2</font></span>)</span>
 ;<br/><a NAME="E5:192_1_1_2_4_1_3"/><b>then </b>
<font color="Maroon" title="c12">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="c15">s2</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/><b>then </b><i><font color="Green" title="E201">A80</font></i>: <font color="Maroon" title="c17">H1'</font> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</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_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a>, <a class="txt" href="group_9.html#E7:192_1_1_2_4"><i><font color="Green" title="E196">A63</font></i></a>, <a class="txt" href="group_9.html#E1:192_1_1_2_4_1_3"><i><font color="Green" title="E200">A79</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">N2</font>
<b>by </b><i><a class="txt" href="group_9.html#E11:192_1"><i><font color="Green" title="E174">A10</font></i></a>, <a class="ref" href="finseq_5.html#T76" title="FINSEQ_5:th.76">FINSEQ_5:76</a></i>
;<br/><a NAME="E7:192_1_1_2_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c3">s1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E15:192_1"><i><font color="Green" title="E177">A13</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:192_1_1_2_4_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c15">s2</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a>, <a class="txt" href="group_9.html#E14:192_1"><i><font color="Green" title="E176">A12</font></i></a></i>;<br/><a NAME="E9:192_1_1_2_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c15">s2</font></span>)</span>
 ;<br/><a NAME="E10:192_1_1_2_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c15">s2</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/><b>then </b><i><font color="Green" title="E202">A81</font></i>: <font color="Maroon" title="c18">N1'</font> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a>, <a class="txt" href="group_9.html#E1:192_1_1_2_4_1_3"><i><font color="Green" title="E200">A79</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_4"><i><font color="Green" title="E197">A64</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</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#E15:192_1"><i><font color="Green" title="E177">A13</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon" title="c9">N1</font>
<b>by </b><i><a class="txt" href="group_9.html#E17:192_1"><i><font color="Green" title="E178">A14</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="192_1_1_2_4_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:192_1_1_2_4_1_3_3"/>
<font color="Maroon" title="c17">H1'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c18">N1'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/><a NAME="E2:192_1_1_2_4_1_3_3"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c9">N1</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2"><i><font color="Green" title="E181">A18</font></i></a>, <a class="txt" href="group_9.html#E6:192_1_1_2_4_1_3"><i><font color="Green" title="E201">A80</font></i></a>, <a class="txt" href="group_9.html#E11:192_1_1_2_4_1_3"><i><font color="Green" title="E202">A81</font></i></a>, <a class="ref" href="group_9.html#T76" target="_self" title="GROUP_9:th.76">Th76</a></i>;<br/><b>hence </b><a NAME="E3:192_1_1_2_4_1_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2"><i><font color="Green" title="E180">A17</font></i></a>, <a class="ref" href="group_9.html#T81" target="_self" title="GROUP_9:th.81">Th81</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:192_1_1_2_4_1_3"/>
not <font color="Maroon" title="c17">H1'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c18">N1'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_1_1_2_4_1_4"><b>suppose </b></a><a NAME="E1:192_1_1_2_4_1_4"/><i><font color="Green" title="E200">A82</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 ;<br/><div class="add"><a NAME="E2:192_1_1_2_4_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c16">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:192_1_1_2_4_1_4"/><b>then </b><i><font color="Green" title="E201">A83</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>set </b><font color="Maroon" title="c19">j'</font> = <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><a NAME="E4:192_1_1_2_4_1_4"/><i><font color="Green" title="E202">A84</font></i>: 
0 <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="c12">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="E5:192_1_1_2_4_1_4"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1_1_2_4_1_4"><i><font color="Green" title="E200">A82</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:192_1_1_2_4_1_4"/><b>then </b><i><font color="Green" title="E203">A85</font></i>: 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">j'</font> = <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="E8:192_1_1_2_4_1_4"/><i><font color="Green" title="E204">A86</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c15">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2_4"><i><font color="Green" title="E193">A60</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:192_1_1_2_4_1_4"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">j'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">j'</font>
 <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="E10:192_1_1_2_4_1_4"/><b>then </b><i><font color="Green" title="E205">A87</font></i>: 
<font color="Maroon" title="c12">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> <font color="Maroon" title="c20">j'</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#E3:192_1_1_2_4_1_4"><i><font color="Green" title="E201">A83</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:192_1_1_2_4_1_4"/><i><font color="Green" title="E206">A88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c15">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2_4"><i><font color="Green" title="E195">A62</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:192_1_1_2_4_1_4"/>
0 <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="c20">j'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:192_1_1_2_4_1_4"><i><font color="Green" title="E203">A85</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:192_1_1_2_4_1_4"/><b>then </b>
<font color="Maroon" title="c20">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="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E8:192_1_1_2_4_1_4"><i><font color="Green" title="E204">A86</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a></i>;<br/><a NAME="E14:192_1_1_2_4_1_4"/><b>then </b><i><font color="Green" title="E207">A89</font></i>: 
<font color="Maroon" title="c20">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="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="E15:192_1_1_2_4_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j'</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#E4:192_1_1_2_4_1_4"><i><font color="Green" title="E202">A84</font></i></a>, <a class="txt" href="group_9.html#E1:192_1_1_2_4_1_4"><i><font color="Green" title="E200">A82</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:192_1_1_2_4_1_4"/><b>then </b>
<font color="Maroon" title="c20">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="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E11:192_1_1_2_4_1_4"><i><font color="Green" title="E206">A88</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a></i>;<br/><a NAME="E17:192_1_1_2_4_1_4"/><b>then </b><i><font color="Green" title="E208">A90</font></i>: 
<font color="Maroon" title="c20">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="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/><i><font color="Green" title="E209">A91</font></i>: <font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j'</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#E1:192_1_1_2_4"><i><font color="Green" title="E192">A59</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">j'</font>
<b>by </b><i><a class="txt" href="group_9.html#E3:192_1_1_2_4_1_4"><i><font color="Green" title="E201">A83</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_4_1_4"><i><font color="Green" title="E204">A86</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j'</font>
<b>by </b><i><a class="txt" href="group_9.html#E14:192_1_1_2_4_1_4"><i><font color="Green" title="E207">A89</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><font color="Maroon" title="c15">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c15">s2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">j'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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#E4:192_1_1_2_4"><i><font color="Green" title="E194">A61</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">j'</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#E10:192_1_1_2_4_1_4"><i><font color="Green" title="E205">A87</font></i></a>, <a class="txt" href="group_9.html#E11:192_1_1_2_4_1_4"><i><font color="Green" title="E206">A88</font></i></a>, <a class="txt" href="group_9.html#E23:192_1_1_2"><i><font color="Green" title="E190">A27</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<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="c20">j'</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#E17:192_1_1_2_4_1_4"><i><font color="Green" title="E208">A90</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E20:192_1_1_2_4_1_4"/>
not <font color="Maroon" title="c17">H1'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c18">N1'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E18:192_1_1_2_4_1_4"><i><font color="Green" title="E209">A91</font></i></a>, <a class="txt" href="group_9.html#E7:192_1_1_2_4"><i><font color="Green" title="E196">A63</font></i></a>, <a class="txt" href="group_9.html#E8:192_1_1_2_4"><i><font color="Green" title="E197">A64</font></i></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a>, <a class="txt" href="group_9.html#E14:192_1_1_2_4_1_4"><i><font color="Green" title="E207">A89</font></i></a>, <a class="txt" href="group_9.html#E17:192_1_1_2_4_1_4"><i><font color="Green" title="E208">A90</font></i></a>, <a class="txt" href="group_9.html#E12:192_1_1_2_4"><i><font color="Green" title="E199">A66</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E28:192_1_1_2"/><b>then </b><i><font color="Green" title="E192">A92</font></i>: 
<font color="Maroon" title="c15">s2</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a>
 <b>by </b><i><a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a></i>;<br/><a NAME="E29:192_1_1_2"/><i><font color="Green" title="E193">A93</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">s2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c15">s2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E30:192_1_1_2"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c15">s2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</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#E13:192_1"><i><font color="Green" title="E175">A11</font></i></a>, <a class="txt" href="group_9.html#E190"><i><font color="Green" title="E163">Lm44</font></i></a></i>;<br/><a NAME="E31:192_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">s2</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E29:192_1_1_2"><i><font color="Green" title="E193">A93</font></i></a>, <a class="ref" href="group_9.html#D32" target="_self" title="GROUP_9:def.32">Def32</a></i>;<br/><b>hence </b><a NAME="E32:192_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E1:192_1"><i><font color="Green" title="E167">A3</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a>, <a class="txt" href="group_9.html#E24:192_1_1_2"><i><font color="Green" title="E191">A28</font></i></a>, <a class="txt" href="group_9.html#E28:192_1_1_2"><i><font color="Green" title="E192">A92</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="192_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:192_2"/><i><font color="Green" title="E167">A94</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   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> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>
 ;<br/><b>assume </b><a NAME="E2:192_2"/><i><font color="Green" title="E168">A95</font></i>: 
not <font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:192_2_1"/>
(  not <font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a> or  ex <font color="Olive" title="b1">s2</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">s2</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a> &amp; <font color="Olive" title="b1">s2</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> ) )
 </span><b>by </b><i><a class="txt" href="group_9.html#E2:192_2"><i><font color="Green" title="E168">A95</font></i></a>, <a class="ref" href="group_9.html#D34" target="_self" title="GROUP_9:def.34">Def34</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_2_1_1"><b>suppose </b></a><a NAME="E1:192_2_1_1"/>
not <font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a>
 ;<br/><div class="add"><a NAME="E2:192_2_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> &amp;  ex <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">G</font> ex <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="b2">H1</font> st <br/>( <font color="Olive" title="b2">H1</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="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b3">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) )
 <b>by </b><i><a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">i</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> , <font color="Maroon" title="c5">H1</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c6">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="c5">H1</font><b> such that </b><br/><a NAME="E4:192_2_1_1"/><i><font color="Green" title="E169">A96</font></i>: 
( <font color="Maroon" title="c4">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="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> )
 <b>and </b><br/><a NAME="E5:192_2_1_1"/><i><font color="Green" title="E170">A97</font></i>: 
( <font color="Maroon" title="c5">H1</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="c4">i</font> &amp; <font color="Maroon" title="c6">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="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E6:192_2_1_1"/><i><font color="Green" title="E171">A98</font></i>: 
<font color="Maroon" title="c5">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c6">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/><a NAME="E7:192_2_1_1"/>
( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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#E4:192_2_1_1"><i><font color="Green" title="E169">A96</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="E8:192_2_1_1"/><b>then </b><i><font color="Green" title="E172">A99</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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/><a NAME="E9:192_2_1_1"/><b>then </b>
1 <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="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="E10:192_2_1_1"/><b>then </b>
1 <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="txt" href="group_9.html#E8:192_2_1_1"><i><font color="Green" title="E172">A99</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:192_2_1_1"/><b>then </b><i><font color="Green" title="E173">A100</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:192_2_1_1"/><b>then </b>
<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="c3">s1</font></span>)</span></span>)</span> <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="c3">s1</font>
 <b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E13:192_2_1_1"/><b>then </b>
 <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="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E14:192_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a>, <a class="txt" href="group_9.html#E8:192_2_1_1"><i><font color="Green" title="E172">A99</font></i></a></i>;<br/><a NAME="E15:192_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E8:192_2_1_1"><i><font color="Green" title="E172">A99</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:192_2_1_1"/><b>then </b><i><font color="Green" title="E174">A101</font></i>: 
<font color="Maroon" title="c4">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="c3">s1</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="E17:192_2_1_1"/><b>then </b>
( <font color="Maroon" title="c5">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c6">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c6">N1</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E5:192_2_1_1"><i><font color="Green" title="E170">A97</font></i></a>, <a class="txt" href="group_9.html#E6:192_2_1_1"><i><font color="Green" title="E171">A98</font></i></a>, <a class="txt" href="group_9.html#E11:192_2_1_1"><i><font color="Green" title="E173">A100</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E18:192_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c6">N1</font> is  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>
 <b>by </b><i><a class="txt" href="group_9.html#E16:192_2_1_1"><i><font color="Green" title="E174">A101</font></i></a>, <a class="txt" href="group_9.html#E1:192_2"><i><font color="Green" title="E167">A94</font></i></a></i>;<br/><b>hence </b><a NAME="E19:192_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="group_9.html#D13" target="_self" title="GROUP_9:def.13">Def13</a>, <a class="txt" href="group_9.html#E6:192_2_1_1"><i><font color="Green" title="E171">A98</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="192_2_1_2"><b>suppose </b></a><a NAME="E1:192_2_1_2"/>
 ex <font color="Olive" title="b1">s2</font> being  <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">s2</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a> &amp; <font color="Olive" title="b1">s2</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">s2</font> being   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E3:192_2_1_2"/><i><font color="Green" title="E169">A102</font></i>: 
( <font color="Maroon" title="c4">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c4">s2</font> is <a href="group_9.html#V12" title="GROUP_9:attr.12">strictly_decreasing</a> &amp; <font color="Maroon" title="c4">s2</font> <a href="group_9.html#R4" title="GROUP_9:pred.4">is_finer_than</a> <font color="Maroon" title="c3">s1</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:192_2_1_2"/><i><font color="Green" title="E170">A103</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> &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="c4">s2</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">s2</font> &amp; <font color="Maroon" title="c6">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="c4">s2</font> )
 <b>and </b><br/><a NAME="E6:192_2_1_2"/><i><font color="Green" title="E171">A104</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</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="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">s2</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )
 <b>by </b><i><a class="ref" href="group_9.html#T100" target="_self" title="GROUP_9:th.100">Th100</a>, <a class="txt" href="group_9.html#E1:192"><i><font color="Green" title="E165">A1</font></i></a>, <a class="txt" href="group_9.html#E3:192_2_1_2"><i><font color="Green" title="E169">A102</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">H1</font> = <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>, <font color="Maroon" title="c8">H2</font> = <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c9">H</font> = <font color="Maroon" title="c4">s2</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">G</font> <b>by </b><i><a class="txt" href="group_9.html#E5:192_2_1_2"><i><font color="Green" title="E170">A103</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">H1</font> = <font color="Maroon" title="c7">H1</font>, <font color="Maroon" title="c11">H2</font> = <font color="Maroon" title="c8">H2</font>, <font color="Maroon" title="c12">H</font> = <font color="Maroon" title="c9">H</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">H</font> = <font color="Maroon" title="c12">H</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c10">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E5:192_2_1_2"><i><font color="Green" title="E170">A103</font></i></a>, <a class="txt" href="group_9.html#E6:192_2_1_2"><i><font color="Green" title="E171">A104</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">H2</font> = <font color="Maroon" title="c11">H2</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c10">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E5:192_2_1_2"><i><font color="Green" title="E170">A103</font></i></a>, <a class="ref" href="group_9.html#D31" target="_self" title="GROUP_9:def.31">Def31</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">H3</font> =  <a href="group_9.html#G1" title="GROUP_9:aggr.1">HGrWOpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H2</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c14">H2</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c14">H2</font> #) as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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">H</font> <b>by </b><i><a class="ref" href="group_9.html#T40" target="_self" title="GROUP_9:th.40">Th40</a>, <a class="txt" href="group_9.html#E5:192_2_1_2"><i><font color="Green" title="E170">A103</font></i></a>, <a class="txt" href="group_9.html#E6:192_2_1_2"><i><font color="Green" title="E171">A104</font></i></a>, <a class="ref" href="group_9.html#T101" target="_self" title="GROUP_9:th.101">Th101</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">H2'</font> = <font color="Maroon" title="c14">H2</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="c13">H</font> <b>by </b><i><a class="txt" href="group_9.html#E5:192_2_1_2"><i><font color="Green" title="E170">A103</font></i></a>, <a class="txt" href="group_9.html#E6:192_2_1_2"><i><font color="Green" title="E171">A104</font></i></a>, <a class="ref" href="group_9.html#T101" target="_self" title="GROUP_9:th.101">Th101</a>, <a class="ref" href="group_9.html#T40" target="_self" title="GROUP_9:th.40">Th40</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">J</font> = <font color="Maroon" title="c13">H</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c16">H2'</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font> <b>by </b><i><a class="ref" href="group_9.html#T44" target="_self" title="GROUP_9:th.44">Th44</a></i>;<br/><div><i><font color="Green" title="E172">A105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="192_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:192_2_1_2_1"/>
<font color="Maroon" title="c17">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font></span>)</span>
 ;<br/><a NAME="E2:192_2_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c14">H2</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_9.html#T22" target="_self" title="GROUP_9:th.22">Th22</a></i>;<br/><a NAME="E3:192_2_1_2_1"/><b>then </b><i><font color="Green" title="E173">A106</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H1</font>
 <b>by </b><i><a class="ref" href="group_9.html#T22" target="_self" title="GROUP_9:th.22">Th22</a></i>;<br/><a NAME="E4:192_2_1_2_1"/><b>then </b><i><font color="Green" title="E174">A107</font></i>: 
<font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">H1</font> = <font color="Maroon" title="c10">H1</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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">H</font> ;<br/><a NAME="E6:192_2_1_2_1"/>
<font color="Maroon" title="c18">H1</font> <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="c13">H</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:192_2_1_2_1"><i><font color="Green" title="E173">A106</font></i></a>, <a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/><a NAME="E7:192_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">H</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c18">H1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="ref" href="group_9.html#T57" target="_self" title="GROUP_9:th.57">Th57</a></i>;<br/><b>hence </b><a NAME="E8:192_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E4:192_2_1_2_1"><i><font color="Green" title="E174">A107</font></i></a>, <a class="txt" href="group_9.html#E3:192_2_1_2"><i><font color="Green" title="E169">A102</font></i></a>, <a class="txt" href="group_9.html#E5:192_2_1_2"><i><font color="Green" title="E170">A103</font></i></a>, <a class="ref" href="group_9.html#D33" target="_self" title="GROUP_9:def.33">Def33</a>, <a class="txt" href="group_9.html#E6:192_2_1_2"><i><font color="Green" title="E171">A104</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="192_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:192_2_1_2_2"/>
<font color="Maroon" title="c17">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font></span>)</span>
 ;<br/><a NAME="E2:192_2_1_2_2"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c15">H3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></i>;<br/><a NAME="E3:192_2_1_2_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="group_9.html#T22" target="_self" title="GROUP_9:th.22">Th22</a></i>;<br/><a NAME="E4:192_2_1_2_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/><a NAME="E5:192_2_1_2_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c14">H2</font>
 <b>by </b><i><a class="ref" href="group_9.html#T43" target="_self" title="GROUP_9:th.43">Th43</a></i>;<br/><b>hence </b><a NAME="E6:192_2_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E6:192_2_1_2"><i><font color="Green" title="E171">A104</font></i></a>, <a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E16:192_2_1_2"/><b>then </b><i><font color="Green" title="E173">A108</font></i>: 
<font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font> is  not  <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>
 <b>by </b><i><a class="txt" href="group_9.html#E14:192_2_1_2"><i><font color="Green" title="E172">A105</font></i></a>, <a class="ref" href="group_9.html#D13" target="_self" title="GROUP_9:def.13">Def13</a></i>;<br/><a NAME="E17:192_2_1_2"/>
( <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> &amp; <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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#E5:192_2_1_2"><i><font color="Green" title="E170">A103</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="E18:192_2_1_2"/><b>then </b><i><font color="Green" title="E174">A109</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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/><a NAME="E19:192_2_1_2"/><b>then </b>
1 <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="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="E20:192_2_1_2"/><b>then </b>
1 <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="txt" href="group_9.html#E18:192_2_1_2"><i><font color="Green" title="E174">A109</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E21:192_2_1_2"/><b>then </b><i><font color="Green" title="E175">A110</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:192_2_1_2"/><b>then </b>
<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="c3">s1</font></span>)</span></span>)</span> <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="c3">s1</font>
 <b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E23:192_2_1_2"/><b>then </b>
 <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="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E24:192_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a>, <a class="txt" href="group_9.html#E18:192_2_1_2"><i><font color="Green" title="E174">A109</font></i></a></i>;<br/><a NAME="E25:192_2_1_2"/><b>then </b>
<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> <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="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E18:192_2_1_2"><i><font color="Green" title="E174">A109</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E26:192_2_1_2"/><b>then </b><i><font color="Green" title="E176">A111</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> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <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="E27:192_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <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="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c14">H2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E21:192_2_1_2"><i><font color="Green" title="E175">A110</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>hence </b><a NAME="E28:192_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E16:192_2_1_2"><i><font color="Green" title="E173">A108</font></i></a>, <a class="txt" href="group_9.html#E1:192_2"><i><font color="Green" title="E167">A94</font></i></a>, <a class="txt" href="group_9.html#E26:192_2_1_2"><i><font color="Green" title="E176">A111</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:192"/>
( <font color="Maroon" title="c3">s1</font> is <a href="group_9.html#V13" title="GROUP_9:attr.13">jordan_holder</a> iff for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   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> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E2:192"><i><font color="Green" title="E166">A2</font></i></a></i>;<br/>


</div>
