<?xml version="1.0"?>
<div class="add">

<a NAME="E1:196_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">l99</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<a NAME="E3:196_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E4:196_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A3</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:196_1_1"/>
<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 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E169">A1</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">l9</font> = <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 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b3">H1</font>, <font color="Olive" title="b4">H2</font>, <font color="Olive" title="b5">H3</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b3">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> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">H2</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="b5">H3</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="Olive" title="b2">j</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Olive" title="b5">H3</font></span>)</span> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) );<br/>
<a NAME="E8:196_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E9:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A4</font></span>: 
<font color="Maroon" title="c5">l99</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/>
<a NAME="E10:196_1_1"/>
<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#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E169">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E11:196_1_1"/><span class="kw">then </span>
 <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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A5</font></span>: 
<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 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E13:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:196_1_1"><span class="lab"><font color="Green" title="E171">A3</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">l</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:196_1_1"><span class="lab"><font color="Green" title="E173">A5</font></span></a>, <a class="txt" href="#E5:196_1_1"><span class="lab"><font color="Green" title="E171">A3</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<a NAME="E15:196_1_1"/><span class="lab"><font color="Green" title="E175">A7</font></span>: 
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> <font color="Maroon" title="c7">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1"><span class="lab"><font color="Green" title="E174">A6</font></span></a></span>;<br/>
<a NAME="E16:196_1_1"/><span class="lab"><font color="Green" title="E176">A8</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">j</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> or  not <font color="Olive" title="b2">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> or  not <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> or  not <font color="Olive" title="b1">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> or  not <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:196_1_1_1"/><span class="lab"><font color="Green" title="E177">A9</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> or  not <font color="Olive" title="b1">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> or  not <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E2:196_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:196_1_1_1"/><span class="lab"><font color="Green" title="E178">A10</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">and </span><br/><a NAME="E5:196_1_1_1"/><span class="lab"><font color="Green" title="E179">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E6:196_1_1_1"/><span class="lab"><font color="Green" title="E180">A12</font></span>: 
<font color="Maroon" title="c9">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E7:196_1_1_1"/><span class="lab"><font color="Green" title="E181">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">and </span><br/><a NAME="E8:196_1_1_1"/><span class="lab"><font color="Green" title="E182">A14</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">i9</font> = <font color="Maroon" title="c9">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<a NAME="E9:196_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:196_1_1_1"><span class="lab"><font color="Green" title="E179">A11</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i9</font> = <font color="Maroon" title="c9">i</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<span class="lab"><font color="Green" title="E183">A15</font></span>: 1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_1"><span class="lab"><font color="Green" title="E177">A9</font></span></a>, <a class="txt" href="#E4:196_1_1_1"><span class="lab"><font color="Green" title="E178">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">j</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></span>
;<br/>
<a NAME="E12:196_1_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:196_1_1_1_2_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l99</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l99</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E184">A16</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:196_1_1_1_2_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:196_1_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_1"><span class="lab"><font color="Green" title="E181">A13</font></span></a>, <a class="txt" href="#E1:196_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E184">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:196_1_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:196_1_1_1"><span class="lab"><font color="Green" title="E183">A15</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:196_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E184">A16</font></span></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:196_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:196_1_1_1"><span class="lab"><font color="Green" title="E182">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:196_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A17</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:196_1_1_1_2_1_2_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l99</font> or 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l99</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_1_2_1_2_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:196_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_1"><span class="lab"><font color="Green" title="E181">A13</font></span></a>, <a class="txt" href="#E8:196_1_1_1"><span class="lab"><font color="Green" title="E182">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E185">A18</font></span>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:196_1_1_1_2_1_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_1"><span class="lab"><font color="Green" title="E181">A13</font></span></a>, <a class="txt" href="#E8:196_1_1_1"><span class="lab"><font color="Green" title="E182">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:196_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E185">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:196_1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:196_1_1_1"><span class="lab"><font color="Green" title="E183">A15</font></span></a>, <a class="txt" href="#E3:196_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E184">A17</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E13:196_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A19</font></span>: 
<font color="Maroon" title="c6">l9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l99</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">l99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_1"><span class="lab"><font color="Green" title="E177">A9</font></span></a>, <a class="txt" href="#E4:196_1_1_1"><span class="lab"><font color="Green" title="E178">A10</font></span></a>, <a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>;<br/>
<a NAME="E14:196_1_1_1"/>
<font color="Maroon" title="c5">l99</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_1"><span class="lab"><font color="Green" title="E181">A13</font></span></a>, <a class="txt" href="#E8:196_1_1_1"><span class="lab"><font color="Green" title="E182">A14</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/>
<a NAME="E15:196_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A20</font></span>: 
<font color="Maroon" title="c6">l9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1_1"><span class="lab"><font color="Green" title="E184">A19</font></span></a>, <a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>;<br/>
<a NAME="E16:196_1_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="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:196_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:196_1_1_1"><span class="lab"><font color="Green" title="E180">A12</font></span></a>, <a class="txt" href="#E15:196_1_1_1"><span class="lab"><font color="Green" title="E185">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:196_1_1"/><span class="lab"><font color="Green" title="E177">A21</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c7">l</font> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</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="c7">l</font> implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:196_1_1_2"/><span class="lab"><font color="Green" title="E178">A22</font></span>: 
<font color="Maroon" title="c8">k</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="c7">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:196_1_1_2_1"/>
(  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) or <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E169">A1</font></span></a>, <a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="txt" href="#E1:196_1_1_2"><span class="lab"><font color="Green" title="E178">A22</font></span></a>, <a class="txt" href="group_9.html#E194"><span class="lab"><font color="Green" title="E166">Lm45</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_2_1_1"/><span class="lab"><font color="Green" title="E179">A23</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:196_1_1_2_1_1"/><span class="lab"><font color="Green" title="E180">A24</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">and </span><br/><a NAME="E4:196_1_1_2_1_1"/><span class="lab"><font color="Green" title="E181">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E5:196_1_1_2_1_1"/><span class="lab"><font color="Green" title="E182">A26</font></span>: 
<font color="Maroon" title="c9">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E6:196_1_1_2_1_1"/><span class="lab"><font color="Green" title="E183">A27</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">H1</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c12">H2</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c13">H3</font> = <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E181">A25</font></span></a>, <a class="txt" href="#E5:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E182">A26</font></span></a>, <a class="txt" href="#E6:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E183">A27</font></span></a>, <a class="ref" href="group_9.html#T111" target="_self" title="GROUP_9:th.111">Th111</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c11">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c13">H3</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c14">x</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">i1</font>, <font color="Maroon" title="c16">j1</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font>, <font color="Olive" title="b3">H3</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c15">i1</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Olive" title="b3">H3</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="c16">j1</font> implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Olive" title="b3">H3</font></span>)</span> ) &amp; ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">H1</font>, <font color="Maroon" title="c18">H2</font>, <font color="Maroon" title="c19">H3</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c15">i1</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c19">H3</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="c16">j1</font> implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c19">H3</font></span>)</span> ) &amp; ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:196_1_1_2_1_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c15">i1</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c19">H3</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="c16">j1</font> implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c19">H3</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:196_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E184">A28</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">j1</font>
 <span class="kw">and </span><br/><a NAME="E2:196_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E185">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1</font>
 <span class="kw">and </span><br/><a NAME="E3:196_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c15">i1</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E4:196_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E186">A30</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or  not <font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i1</font> or  not <font color="Maroon" title="c19">H3</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="c16">j1</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c19">H3</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E5:196_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E187">A31</font></span>: 
( <font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c19">H3</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="c16">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c19">H3</font></span>)</span></span><br/>

<a NAME="E6:196_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="txt" href="#E3:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E180">A24</font></span></a>, <a class="txt" href="#E4:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E181">A25</font></span></a>, <a class="txt" href="#E6:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E183">A27</font></span></a>, <a class="txt" href="#E1:196_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E184">A28</font></span></a>, <a class="txt" href="#E2:196_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E185">A29</font></span></a>, <a class="txt" href="#E4:196_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E186">A30</font></span></a>, <a class="txt" href="group_9.html#E195"><span class="lab"><font color="Green" title="E167">Lm46</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:196_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c19">H3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E180">A24</font></span></a>, <a class="txt" href="#E1:196_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E184">A28</font></span></a>, <a class="txt" href="#E5:196_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E187">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:196_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span><br/><span class="kw">hence </span><a NAME="E3:196_1_1_2_1_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:196_1_1"><span class="lab"><font color="Green" title="E176">A8</font></span></a>, <a class="txt" href="#E1:196_1_1_2_1_1"><span class="lab"><font color="Green" title="E179">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:196_1_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c14">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_2_1_2"/><span class="lab"><font color="Green" title="E179">A32</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">take </span>
 <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>, <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>]</span><br/><span class="kw">thus </span><a NAME="E2:196_1_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>, <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:196_1_1"><span class="lab"><font color="Green" title="E176">A8</font></span></a>, <a class="txt" href="#E1:196_1_1_2_1_2"><span class="lab"><font color="Green" title="E179">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E19:196_1_1"/><span class="lab"><font color="Green" title="E178">A33</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">l</font> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c7">l</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="#E17:196_1_1"><span class="lab"><font color="Green" title="E177">A21</font></span></a>);<br/></span>
<a NAME="E20:196_1_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> implies <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:196_1_1_3"/><span class="lab"><font color="Green" title="E179">A34</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font></span><br/>

<a NAME="E2:196_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E180">A35</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b3">H1</font>, <font color="Olive" title="b4">H2</font>, <font color="Olive" title="b5">H3</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( ( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b3">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> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">H2</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="b5">H3</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="Olive" title="b2">j</font> implies <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Olive" title="b5">H3</font></span>)</span> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:196_1_1_3_1"/>
(  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E169">A1</font></span></a>, <a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E1:196_1_1_3"><span class="lab"><font color="Green" title="E179">A34</font></span></a>, <a class="txt" href="group_9.html#E194"><span class="lab"><font color="Green" title="E166">Lm45</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_3_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:196_1_1_3_1_1"/><span class="lab"><font color="Green" title="E181">A36</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">and </span><br/><a NAME="E4:196_1_1_3_1_1"/><span class="lab"><font color="Green" title="E182">A37</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">H1</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="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c13">H2</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c14">H3</font> = <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:196_1_1_3_1_1"><span class="lab"><font color="Green" title="E182">A37</font></span></a>, <a class="ref" href="group_9.html#T111" target="_self" title="GROUP_9:th.111">Th111</a></span>;<br/><a NAME="E6:196_1_1_3_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c14">H3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E1:196_1_1_3"><span class="lab"><font color="Green" title="E179">A34</font></span></a>, <a class="txt" href="#E3:196_1_1_3_1_1"><span class="lab"><font color="Green" title="E181">A36</font></span></a>, <a class="txt" href="#E4:196_1_1_3_1_1"><span class="lab"><font color="Green" title="E182">A37</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:196_1_1_3_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_3_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:196_1_1_3_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196_1_1_3"><span class="lab"><font color="Green" title="E180">A35</font></span></a>, <a class="ref" href="group_9.html#D11" target="_self" title="GROUP_9:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_9.html#K6" title="GROUP_9:func.6">the_stable_subgroups_of</a> <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<a NAME="E22:196_1_1"/>
<font color="Maroon" title="c7">l</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="c7">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1"><span class="lab"><font color="Green" title="E174">A6</font></span></a></span>;<br/>
<a NAME="E23:196_1_1"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">l</font>,<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">l</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a></span>;<br/>
<a NAME="E24:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E179">A38</font></span>: 
<font color="Maroon" title="c9">f</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="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E25:196_1_1"/><span class="lab"><font color="Green" title="E180">A39</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">s1</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b3">H</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b3">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s1</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b2">H</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <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="Olive" title="b1">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> holds <br/><font color="Olive" title="b1">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">s1</font> be   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <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="c11">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> holds <br/><font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">H</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <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="c11">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> implies <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> 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="c12">H</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:196_1_1_4"/><span class="lab"><font color="Green" title="E181">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">and </span><br/><a NAME="E2:196_1_1_4"/><span class="lab"><font color="Green" title="E182">A41</font></span>: 
<font color="Maroon" title="c10">i</font> <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="c11">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> or <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> 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="c12">H</font> )</span><br/>

<a NAME="E3:196_1_1_4"/><span class="lab"><font color="Green" title="E183">A42</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E4:196_1_1_4"/><span class="lab"><font color="Green" title="E184">A43</font></span>: 
<font color="Maroon" title="c10">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s1</font></span>)</span> <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196_1_1_4"><span class="lab"><font color="Green" title="E182">A41</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E5:196_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c11">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:196_1_1_4"><span class="lab"><font color="Green" title="E183">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:196_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c11">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_4"><span class="lab"><font color="Green" title="E181">A40</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E7:196_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A44</font></span>: 
<font color="Maroon" title="c10">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="c11">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">H1</font> = <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c14">H2</font> = <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_4"><span class="lab"><font color="Green" title="E181">A40</font></span></a>, <a class="txt" href="#E2:196_1_1_4"><span class="lab"><font color="Green" title="E182">A41</font></span></a>, <a class="ref" href="group_9.html#T111" target="_self" title="GROUP_9:th.111">Th111</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:196_1_1_4"/><span class="lab"><font color="Green" title="E186">A45</font></span>: 
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> 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="c12">H</font></span><br/>

<a NAME="E10:196_1_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_4"><span class="lab"><font color="Green" title="E181">A40</font></span></a>, <a class="txt" href="#E3:196_1_1_4"><span class="lab"><font color="Green" title="E183">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:196_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c11">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:196_1_1_4"><span class="lab"><font color="Green" title="E184">A43</font></span></a></span>;<br/>
<a NAME="E12:196_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c11">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E13:196_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c13">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_4"><span class="lab"><font color="Green" title="E185">A44</font></span></a>, <a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:196_1_1_4"/>
<font color="Maroon" title="c11">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> 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="c12">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:196_1_1_4"><span class="lab"><font color="Green" title="E186">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:196_1_1"/><span class="lab"><font color="Green" title="E181">A46</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c9">f</font> &amp; <font color="Olive" title="b1">k</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="c9">f</font> holds <br/> for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b3">H2</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b2">H1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">k</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="c9">f</font> &amp; <font color="Maroon" title="c10">k</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="c9">f</font> implies  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H2</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:196_1_1_5"/><span class="lab"><font color="Green" title="E182">A47</font></span>: 
<font color="Maroon" title="c10">k</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="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">k</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="c9">f</font> or  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H2</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H1</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">k9</font> = <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">assume </span><a NAME="E2:196_1_1_5"/><span class="lab"><font color="Green" title="E183">A48</font></span>: 
<font color="Maroon" title="c10">k</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="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">H2</font> is   <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H1</font></span><br/>

<a NAME="E3:196_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</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="c7">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:196_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:196_1_1_5"/><span class="lab"><font color="Green" title="E184">A49</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">and </span><br/><a NAME="E7:196_1_1_5"/><span class="lab"><font color="Green" title="E185">A50</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">and </span><br/><a NAME="E8:196_1_1_5"/><span class="lab"><font color="Green" title="E186">A51</font></span>: 
<font color="Maroon" title="c12">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E9:196_1_1_5"/><span class="lab"><font color="Green" title="E187">A52</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">and </span><br/><a NAME="E10:196_1_1_5"/><span class="lab"><font color="Green" title="E188">A53</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E169">A1</font></span></a>, <a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E1:196_1_1_5"><span class="lab"><font color="Green" title="E182">A47</font></span></a>, <a class="txt" href="group_9.html#E194"><span class="lab"><font color="Green" title="E166">Lm45</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">H19</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="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c15">H29</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c16">H39</font> = <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_5"><span class="lab"><font color="Green" title="E185">A50</font></span></a>, <a class="txt" href="#E8:196_1_1_5"><span class="lab"><font color="Green" title="E186">A51</font></span></a>, <a class="txt" href="#E9:196_1_1_5"><span class="lab"><font color="Green" title="E187">A52</font></span></a>, <a class="txt" href="#E10:196_1_1_5"><span class="lab"><font color="Green" title="E188">A53</font></span></a>, <a class="ref" href="group_9.html#T111" target="_self" title="GROUP_9:th.111">Th111</a></span>;<br/>
<a NAME="E12:196_1_1_5"/><span class="lab"><font color="Green" title="E189">A54</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H19</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">H29</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c16">H39</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E1:196_1_1_5"><span class="lab"><font color="Green" title="E182">A47</font></span></a>, <a class="txt" href="#E6:196_1_1_5"><span class="lab"><font color="Green" title="E184">A49</font></span></a>, <a class="txt" href="#E7:196_1_1_5"><span class="lab"><font color="Green" title="E185">A50</font></span></a>, <a class="txt" href="#E8:196_1_1_5"><span class="lab"><font color="Green" title="E186">A51</font></span></a>, <a class="txt" href="#E9:196_1_1_5"><span class="lab"><font color="Green" title="E187">A52</font></span></a>, <a class="txt" href="#E10:196_1_1_5"><span class="lab"><font color="Green" title="E188">A53</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">H1</font>, <font color="Maroon" title="c18">H2</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c18">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="c17">H1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E13:196_1_1_5"/><span class="lab"><font color="Green" title="E190">A55</font></span>: 
<font color="Maroon" title="c17">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c18">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="c17">H1</font> )</span><br/>

<a NAME="E14:196_1_1_5"/><span class="lab"><font color="Green" title="E191">A56</font></span>: 
<font color="Maroon" title="c14">H19</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="c15">H29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:196_1_1"><span class="lab"><font color="Green" title="E180">A39</font></span></a>, <a class="txt" href="#E7:196_1_1_5"><span class="lab"><font color="Green" title="E185">A50</font></span></a>, <a class="txt" href="#E8:196_1_1_5"><span class="lab"><font color="Green" title="E186">A51</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E15:196_1_1_5"/><span class="lab"><font color="Green" title="E192">A57</font></span>: 
<font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">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="c17">H1</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:196_1_1_5_1"/>
( <font color="Maroon" title="c13">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_5_1_1"/><span class="lab"><font color="Green" title="E193">A58</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">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="c17">H1</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c19">j9</font> = <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E3:196_1_1_5_1_1"/>
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:196_1_1_5"><span class="lab"><font color="Green" title="E188">A53</font></span></a>, <a class="txt" href="#E1:196_1_1_5_1_1"><span class="lab"><font color="Green" title="E193">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:196_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E194">A59</font></span>: 
<font color="Maroon" title="c19">j9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">H399</font> = <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:196_1_1_5"><span class="lab"><font color="Green" title="E187">A52</font></span></a>, <a class="txt" href="#E10:196_1_1_5"><span class="lab"><font color="Green" title="E188">A53</font></span></a>, <a class="ref" href="group_9.html#T111" target="_self" title="GROUP_9:th.111">Th111</a></span>;<br/><a NAME="E6:196_1_1_5_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:196_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E195">A60</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:196_1_1_5"><span class="lab"><font color="Green" title="E187">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:196_1_1_5_1_1"/><span class="lab"><font color="Green" title="E196">A61</font></span>: 
<font color="Maroon" title="c20">H399</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">H39</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:196_1_1"><span class="lab"><font color="Green" title="E180">A39</font></span></a>, <a class="txt" href="#E9:196_1_1_5"><span class="lab"><font color="Green" title="E187">A52</font></span></a>, <a class="txt" href="#E10:196_1_1_5"><span class="lab"><font color="Green" title="E188">A53</font></span></a></span>;<br/><a NAME="E9:196_1_1_5_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:196_1_1_5"><span class="lab"><font color="Green" title="E184">A49</font></span></a></span>;<br/><a NAME="E10:196_1_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H19</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">H29</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c20">H399</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E2:196_1_1_5"><span class="lab"><font color="Green" title="E183">A48</font></span></a>, <a class="txt" href="#E7:196_1_1_5"><span class="lab"><font color="Green" title="E185">A50</font></span></a>, <a class="txt" href="#E8:196_1_1_5"><span class="lab"><font color="Green" title="E186">A51</font></span></a>, <a class="txt" href="#E15:196_1_1_5"><span class="lab"><font color="Green" title="E192">A57</font></span></a>, <a class="txt" href="#E4:196_1_1_5_1_1"><span class="lab"><font color="Green" title="E194">A59</font></span></a>, <a class="txt" href="#E7:196_1_1_5_1_1"><span class="lab"><font color="Green" title="E195">A60</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:196_1_1_5_1_1"/>
<font color="Maroon" title="c18">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="c17">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1_5"><span class="lab"><font color="Green" title="E190">A55</font></span></a>, <a class="txt" href="#E14:196_1_1_5"><span class="lab"><font color="Green" title="E191">A56</font></span></a>, <a class="txt" href="#E12:196_1_1_5"><span class="lab"><font color="Green" title="E189">A54</font></span></a>, <a class="txt" href="#E8:196_1_1_5_1_1"><span class="lab"><font color="Green" title="E196">A61</font></span></a>, <a class="ref" href="group_9.html#T92" target="_self" title="GROUP_9:th.92">Th92</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_5_1_2"/><span class="lab"><font color="Green" title="E193">A62</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">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="c17">H1</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_5_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:196_1_1_5_1_2_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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 or <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_5_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_5_1_2_1_1"/><span class="lab"><font color="Green" title="E194">A63</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">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="c17">H1</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c19">i9</font> = <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E2:196_1_1_5_1_2_1_1"/><span class="lab"><font color="Green" title="E195">A64</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">j9</font> = 1;<br/><a NAME="E3:196_1_1_5_1_2_1_1"/>
<font color="Maroon" title="c14">H19</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c17">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1_5"><span class="lab"><font color="Green" title="E190">A55</font></span></a>, <a class="txt" href="#E12:196_1_1_5"><span class="lab"><font color="Green" title="E189">A54</font></span></a>, <a class="ref" href="group_9.html#T35" target="_self" title="GROUP_9:th.35">Th35</a></span>;<br/><a NAME="E4:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">H19</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c17">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/><a NAME="E5:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E196">A65</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><a NAME="E6:196_1_1_5_1_2_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_5"><span class="lab"><font color="Green" title="E185">A50</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E197">A66</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:196_1_1_5_1_2_1_1"/>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">l9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:196_1_1_5"><span class="lab"><font color="Green" title="E186">A51</font></span></a>, <a class="txt" href="#E1:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E194">A63</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E9:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E198">A67</font></span>: 
<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="c6">l9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E10:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E199">A68</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><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="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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E11:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
<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="c3">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E195">A64</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
<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="c3">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E197">A66</font></span></a></span>;<br/><a NAME="E13:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E200">A69</font></span>: 
<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="c3">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E14:196_1_1_5_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E15:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E201">A70</font></span>: 
<font color="Maroon" title="c5">l99</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">H199</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"><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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c22">H299</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="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c23">H399</font> = <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E198">A67</font></span></a>, <a class="txt" href="#E7:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E197">A66</font></span></a>, <a class="ref" href="group_9.html#T111" target="_self" title="GROUP_9:th.111">Th111</a></span>;<br/><a NAME="E17:196_1_1_5_1_2_1_1"/>
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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E197">A66</font></span></a>, <a class="txt" href="#E2:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E195">A64</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
<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#K2" title="NAT_1: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E199">A68</font></span></a></span>;<br/><a NAME="E19:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
<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#K2" title="NAT_1: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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E20:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E202">A71</font></span>: 
<font color="Maroon" title="c21">H199</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="c22">H299</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E200">A69</font></span></a>, <a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_5_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c22">H299</font> implies <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/><a NAME="E1:196_1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c22">H299</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/><a NAME="E2:196_1_1_5_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E203">A72</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c22">H299</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E3:196_1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c22">H299</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/><span class="kw">hence </span><a NAME="E4:196_1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196_1_1_5_1_2_1_1_1"><span class="lab"><font color="Green" title="E203">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E22:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c22">H299</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E23:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E203">A73</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c22">H299</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="c22">H299</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E24:196_1_1_5_1_2_1_1"/><span class="lab"><font color="Green" title="E204">A74</font></span>: 
<font color="Maroon" title="c23">H399</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="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/><a NAME="E25:196_1_1_5_1_2_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:196_1_1_5"><span class="lab"><font color="Green" title="E184">A49</font></span></a>, <a class="txt" href="#E1:196_1_1_5_1_2"><span class="lab"><font color="Green" title="E193">A62</font></span></a></span>;<br/><a NAME="E26:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H199</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H299</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c23">H399</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E2:196_1_1_5"><span class="lab"><font color="Green" title="E183">A48</font></span></a>, <a class="txt" href="#E15:196_1_1_5"><span class="lab"><font color="Green" title="E192">A57</font></span></a>, <a class="txt" href="#E9:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E198">A67</font></span></a>, <a class="txt" href="#E7:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E197">A66</font></span></a>, <a class="txt" href="#E15:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E201">A70</font></span></a></span>;<br/><a NAME="E27:196_1_1_5_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H199</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c22">H299</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E204">A74</font></span></a>, <a class="txt" href="#E23:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E203">A73</font></span></a>, <a class="ref" href="group_9.html#T18" target="_self" title="GROUP_9:th.18">Th18</a></span>;<br/><a NAME="E28:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E205">A75</font></span>: 
<font color="Maroon" title="c18">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E202">A71</font></span></a>, <a class="ref" href="group_9.html#T36" target="_self" title="GROUP_9:th.36">Th36</a></span>;<br/><a NAME="E29:196_1_1_5_1_2_1_1"/>
<font color="Maroon" title="c15">H29</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c16">H39</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c15">H29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E101"><span class="lab"><font color="Green" title="E90">Lm34</font></span></a></span>;<br/><a NAME="E30:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E206">A76</font></span>: 
<font color="Maroon" title="c17">H1</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c15">H29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1_5"><span class="lab"><font color="Green" title="E190">A55</font></span></a>, <a class="txt" href="#E14:196_1_1_5"><span class="lab"><font color="Green" title="E191">A56</font></span></a>, <a class="txt" href="#E12:196_1_1_5"><span class="lab"><font color="Green" title="E189">A54</font></span></a>, <a class="ref" href="group_9.html#T37" target="_self" title="GROUP_9:th.37">Th37</a></span>;<br/><a NAME="E31:196_1_1_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E207">A77</font></span>: 
<font color="Maroon" title="c17">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c15">H29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_5_1_2_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">H9</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c17">H1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">H9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">H2</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c18">H2</font> #) implies <font color="Maroon" title="c24">H9</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:196_1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E208">A78</font></span>: 
<font color="Maroon" title="c24">H9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">H2</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c18">H2</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">H9</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_5_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">H1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c24">H9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">H9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c25">a</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c26">a9</font> = <font color="Maroon" title="c25">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">H29</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E206">A76</font></span></a>, <a class="ref" href="group_9.html#T2" target="_self" title="GROUP_9:th.2">Th2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_5_1_2_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c27">H1s9</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H19</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c14">H19</font> #) as   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c15">H29</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:196_1_1_5"><span class="lab"><font color="Green" title="E191">A56</font></span></a>, <a class="txt" href="group_9.html#E20"><span class="lab"><font color="Green" title="E19">Lm7</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c24">H9</font> implies <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">H9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c25">a</font> )</span><br/><span class="kw">assume </span><a NAME="E2:196_1_1_5_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c24">H9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">H9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c25">a</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">H1</font><span class="kw"> such that </span><br/><a NAME="E4:196_1_1_5_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E209">A79</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c29">b</font>
 <span class="kw">and </span><br/><a NAME="E5:196_1_1_5_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E210">A80</font></span>: 
<font color="Maroon" title="c29">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c24">H9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T103" title="GROUP_2:th.103">GROUP_2:103</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c30">b9</font> = <font color="Maroon" title="c29">b</font>;<br/><a NAME="E6:196_1_1_5_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E211">A81</font></span>: 
<font color="Maroon" title="c17">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c15">H29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E206">A76</font></span></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">b9</font> = <font color="Maroon" title="c29">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">H29</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T42" title="GROUP_2:th.42">GROUP_2:42</a></span>;<br/><a NAME="E8:196_1_1_5_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c31">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:196_1_1_5_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E209">A79</font></span></a>, <a class="txt" href="#E6:196_1_1_5_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E211">A81</font></span></a>, <a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</a></span>;<br/><a NAME="E9:196_1_1_5_1_2_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c26">a9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c27">H1s9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">H1s9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c26">a9</font> &amp; <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">a9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c27">H1s9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E205">A75</font></span></a>, <a class="txt" href="#E1:196_1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E208">A78</font></span></a>, <a class="txt" href="#E5:196_1_1_5_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E210">A80</font></span></a>, <a class="ref" href="group_2.html#T103" title="GROUP_2:th.103">GROUP_2:103</a>, <a class="ref" href="group_3.html#T118" title="GROUP_3:th.118">GROUP_3:118</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">b99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">H29</font><span class="kw"> such that </span><br/><a NAME="E11:196_1_1_5_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E212">A82</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">b99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c26">a9</font>
 <span class="kw">and </span><br/><a NAME="E12:196_1_1_5_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E213">A83</font></span>: 
<font color="Maroon" title="c32">b99</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c27">H1s9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T104" title="GROUP_2:th.104">GROUP_2:104</a></span>;<br/><a NAME="E13:196_1_1_5_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c32">b99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:196_1_1_5_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E213">A83</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">b99</font> = <font color="Maroon" title="c32">b99</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">H1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E196">A65</font></span></a></span>;<br/><a NAME="E15:196_1_1_5_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">b99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E207">A77</font></span></a>, <a class="txt" href="#E11:196_1_1_5_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E212">A82</font></span></a>, <a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</a></span>;<br/><span class="kw">hence </span><a NAME="E16:196_1_1_5_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">H9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E205">A75</font></span></a>, <a class="txt" href="#E1:196_1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E208">A78</font></span></a>, <a class="txt" href="#E12:196_1_1_5_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E213">A83</font></span></a>, <a class="ref" href="group_2.html#T104" title="GROUP_2:th.104">GROUP_2:104</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:196_1_1_5_1_2_1_1_2_1"/>
<font color="Maroon" title="c25">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c24">H9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">H9</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:196_1_1_5_1_2_1_1_2"/>
<font color="Maroon" title="c24">H9</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T118" title="GROUP_3:th.118">GROUP_3:118</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E33:196_1_1_5_1_2_1_1"/>
<font color="Maroon" title="c18">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="c17">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1_5"><span class="lab"><font color="Green" title="E190">A55</font></span></a>, <a class="txt" href="#E12:196_1_1_5"><span class="lab"><font color="Green" title="E189">A54</font></span></a>, <a class="txt" href="#E28:196_1_1_5_1_2_1_1"><span class="lab"><font color="Green" title="E205">A75</font></span></a>, <a class="ref" href="group_9.html#D10" target="_self" title="GROUP_9:def.10">Def10</a>, <a class="ref" href="group_9.html#T35" target="_self" title="GROUP_9:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_5_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_5_1_2_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">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="c17">H1</font></span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c18">H2</font> = 
 <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E2:196_1_1_5"><span class="lab"><font color="Green" title="E183">A48</font></span></a>, <a class="txt" href="#E6:196_1_1_5"><span class="lab"><font color="Green" title="E184">A49</font></span></a>, <a class="txt" href="#E15:196_1_1_5"><span class="lab"><font color="Green" title="E192">A57</font></span></a>, <a class="txt" href="#E1:196_1_1_5_1_2"><span class="lab"><font color="Green" title="E193">A62</font></span></a></span>
<br/>.= 
 <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c17">H1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#T15" target="_self" title="GROUP_9:th.15">Th15</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:196_1_1_5_1_2_1_2"/>
<font color="Maroon" title="c18">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="c17">H1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E27:196_1_1"/>
( <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 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E169">A1</font></span></a>, <a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E28:196_1_1"/><span class="kw">then </span>
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E29:196_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">l</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E31:196_1_1"/><span class="lab"><font color="Green" title="E182">A84</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">and </span><br/><a NAME="E32:196_1_1"/><span class="lab"><font color="Green" title="E183">A85</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">and </span><br/><a NAME="E33:196_1_1"/><span class="lab"><font color="Green" title="E184">A86</font></span>: 
<font color="Maroon" title="c10">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E34:196_1_1"/><span class="lab"><font color="Green" title="E185">A87</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">and </span><br/><a NAME="E35:196_1_1"/><span class="lab"><font color="Green" title="E186">A88</font></span>: 
<font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E169">A1</font></span></a>, <a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E170">A2</font></span></a>, <a class="txt" href="#E15:196_1_1"><span class="lab"><font color="Green" title="E175">A7</font></span></a>, <a class="txt" href="group_9.html#E194"><span class="lab"><font color="Green" title="E166">Lm45</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">i9</font> = <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<a NAME="E36:196_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:196_1_1"><span class="lab"><font color="Green" title="E183">A85</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">i9</font> = <font color="Maroon" title="c10">i</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">H1</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="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c15">H2</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c16">H3</font> = <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:196_1_1"><span class="lab"><font color="Green" title="E183">A85</font></span></a>, <a class="txt" href="#E33:196_1_1"><span class="lab"><font color="Green" title="E184">A86</font></span></a>, <a class="txt" href="#E34:196_1_1"><span class="lab"><font color="Green" title="E185">A87</font></span></a>, <a class="txt" href="#E35:196_1_1"><span class="lab"><font color="Green" title="E186">A88</font></span></a>, <a class="ref" href="group_9.html#T111" target="_self" title="GROUP_9:th.111">Th111</a></span>;<br/>
<a NAME="E39:196_1_1"/>
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:196_1_1"><span class="lab"><font color="Green" title="E182">A84</font></span></a></span>;<br/>
<a NAME="E40:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E187">A89</font></span>: 
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></span>;<br/>
<a NAME="E41:196_1_1"/><span class="lab"><font color="Green" title="E188">A90</font></span>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_6"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:196_1_1_6_1"/>
( <font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">l99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_6_1_1"/>
<font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:196_1_1_6_1_1"/>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:196_1_1"><span class="lab"><font color="Green" title="E185">A87</font></span></a>, <a class="txt" href="#E35:196_1_1"><span class="lab"><font color="Green" title="E186">A88</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="196_1_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:196_1_1_6_1_2"/>
<font color="Maroon" title="c5">l99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:196_1_1_6_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:196_1_1"><span class="lab"><font color="Green" title="E171">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:196_1_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A91</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:196_1_1"><span class="lab"><font color="Green" title="E187">A89</font></span></a>, <a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></span>;<br/><a NAME="E4:196_1_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>;<br/><a NAME="E5:196_1_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">l99</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:196_1_1"><span class="lab"><font color="Green" title="E186">A88</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:196_1_1_6_1_2"/>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:196_1_1_6_1_2"><span class="lab"><font color="Green" title="E189">A91</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E42:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A92</font></span>: 
<font color="Maroon" title="c16">H3</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="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/>
<a NAME="E43:196_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">i9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">l99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">l99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:196_1_1"><span class="lab"><font color="Green" title="E182">A84</font></span></a>, <a class="txt" href="#E41:196_1_1"><span class="lab"><font color="Green" title="E188">A90</font></span></a></span>;<br/>
<a NAME="E44:196_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">i9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:196_1_1"><span class="lab"><font color="Green" title="E172">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>;<br/>
<a NAME="E45:196_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E190">A93</font></span>: 
<font color="Maroon" title="c15">H2</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="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/>
<a NAME="E46:196_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c16">H3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E15:196_1_1"><span class="lab"><font color="Green" title="E175">A7</font></span></a>, <a class="txt" href="#E31:196_1_1"><span class="lab"><font color="Green" title="E182">A84</font></span></a>, <a class="txt" href="#E32:196_1_1"><span class="lab"><font color="Green" title="E183">A85</font></span></a>, <a class="txt" href="#E33:196_1_1"><span class="lab"><font color="Green" title="E184">A86</font></span></a>, <a class="txt" href="#E34:196_1_1"><span class="lab"><font color="Green" title="E185">A87</font></span></a>, <a class="txt" href="#E35:196_1_1"><span class="lab"><font color="Green" title="E186">A88</font></span></a></span>;<br/>
<a NAME="E47:196_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:196_1_1"><span class="lab"><font color="Green" title="E190">A93</font></span></a>, <a class="txt" href="#E42:196_1_1"><span class="lab"><font color="Green" title="E189">A92</font></span></a>, <a class="ref" href="group_9.html#T19" target="_self" title="GROUP_9:th.19">Th19</a></span>;<br/>
<a NAME="E48:196_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#T34" target="_self" title="GROUP_9:th.34">Th34</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="group_9.html#NM4" title="GROUP_9:NM.4">CompositionSeries</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:196_1_1"><span class="lab"><font color="Green" title="E179">A38</font></span></a>, <a class="txt" href="#E26:196_1_1"><span class="lab"><font color="Green" title="E181">A46</font></span></a>, <a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def31</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b4">H1</font>, <font color="Olive" title="b5">H2</font>, <font color="Olive" title="b6">H3</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b4">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> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b5">H2</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="b2">i</font> &amp; <font color="Olive" title="b6">H3</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="Olive" title="b3">j</font> implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Olive" title="b6">H3</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c18">k</font>, <font color="Maroon" title="c19">i</font>, <font color="Maroon" title="c20">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font>, <font color="Olive" title="b3">H3</font> being    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">H2</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="c19">i</font> &amp; <font color="Olive" title="b3">H3</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="c20">j</font> implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Olive" title="b3">H3</font></span>)</span> ) &amp; ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c21">H1</font>, <font color="Maroon" title="c22">H2</font>, <font color="Maroon" title="c23">H3</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c21">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> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">H2</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="c19">i</font> &amp; <font color="Maroon" title="c23">H3</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="c20">j</font> implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c23">H3</font></span>)</span> ) &amp; ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/>



<a NAME="E50:196_1_1"/><span class="lab"><font color="Green" title="E191">A94</font></span>: 
 for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Olive" title="b1">k</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="c7">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">k</font>, <font color="Maroon" title="c25">i</font>, <font color="Maroon" title="c26">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c26">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i</font> &amp; <font color="Maroon" title="c25">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 implies <font color="Maroon" title="c24">k</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="c7">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:196_1_1_7"/><span class="lab"><font color="Green" title="E192">A95</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c26">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i</font> or  not <font color="Maroon" title="c25">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> or  not <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c24">k</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="c7">l</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:196_1_1_7"/><span class="lab"><font color="Green" title="E193">A96</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i</font>
 <span class="kw">and </span><br/><a NAME="E3:196_1_1_7"/><span class="lab"><font color="Green" title="E194">A97</font></span>: 
<font color="Maroon" title="c25">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> or  not <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c24">k</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="c7">l</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:196_1_1_7"/><span class="lab"><font color="Green" title="E195">A98</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">and </span><br/><a NAME="E5:196_1_1_7"/><span class="lab"><font color="Green" title="E196">A99</font></span>: 
<font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">k</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="c7">l</font></span><br/>

<a NAME="E6:196_1_1_7"/>
<font color="Maroon" title="c25">i</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> <font color="Maroon" title="c6">l9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:196_1_1_7"><span class="lab"><font color="Green" title="E194">A97</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E7:196_1_1_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">l99</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">l99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E8:196_1_1_7"/><span class="kw">then </span>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> &amp; <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l99</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">l99</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_7"><span class="lab"><font color="Green" title="E192">A95</font></span></a>, <a class="txt" href="#E5:196_1_1_7"><span class="lab"><font color="Green" title="E196">A99</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E9:196_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E197">A100</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:196_1_1_7"/>
1 <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="c25">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196_1_1_7"><span class="lab"><font color="Green" title="E193">A96</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E11:196_1_1_7"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c26">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:196_1_1"><span class="lab"><font color="Green" title="E171">A3</font></span></a>, <a class="txt" href="#E4:196_1_1_7"><span class="lab"><font color="Green" title="E195">A98</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:196_1_1_7"/>
<font color="Maroon" title="c24">k</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="c7">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196_1_1_7"><span class="lab"><font color="Green" title="E192">A95</font></span></a>, <a class="txt" href="#E9:196_1_1_7"><span class="lab"><font color="Green" title="E197">A100</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_8"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:196_1_1_8"/><span class="lab"><font color="Green" title="E192">A101</font></span>: 
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:196_1_1_8"/><span class="lab"><font color="Green" title="E193">A102</font></span>: 
( <font color="Maroon" title="c21">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> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">H2</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="c19">i</font> &amp; <font color="Maroon" title="c23">H3</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="c20">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c23">H3</font></span>)</span></span><br/><a NAME="E3:196_1_1_8"/>
<font color="Maroon" title="c18">k</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="c7">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:196_1_1"><span class="lab"><font color="Green" title="E191">A94</font></span></a>, <a class="txt" href="#E1:196_1_1_8"><span class="lab"><font color="Green" title="E192">A101</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:196_1_1_8"/>
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c23">H3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E1:196_1_1_8"><span class="lab"><font color="Green" title="E192">A101</font></span></a>, <a class="txt" href="#E2:196_1_1_8"><span class="lab"><font color="Green" title="E193">A102</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E52:196_1_1"/>
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">i</font> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c21">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> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">H2</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="c19">i</font> &amp; <font color="Maroon" title="c23">H3</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="c20">j</font> implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H2</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c23">H3</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="196_1_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:196_1_1_9"/><span class="lab"><font color="Green" title="E192">A103</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span><br/><a NAME="E2:196_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</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="c7">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:196_1_1"><span class="lab"><font color="Green" title="E174">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:196_1_1_9"/>
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="txt" href="#E1:196_1_1_9"><span class="lab"><font color="Green" title="E192">A103</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E54:196_1_1"/>
( ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:196_1_1"><span class="lab"><font color="Green" title="E178">A33</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
