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

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:62"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">f</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 ;<br/>

<b>assume </b><a NAME="E2:62"/><i><font color="Green">E51</font></i>: 
(  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> or (  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 ;<br/>

<a NAME="E3:62"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="jordan3.html#K5" target="_self">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K4" target="_self">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E4:62"/><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:62"/>
 <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:62"/><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:62"/><i><font color="Green">E63</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1"/>
(  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> or (  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>case </b></a><a NAME="E1:62_1_1_1"/>
 <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:62_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>assume </b><a NAME="E3:62_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E4:62_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E2:62_1_1_1"><i><font color="Green">E65</font></i></a>, , <a class="txt" href="jordan3.html#E10:62"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan3.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E5:62_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan3.html#E2:62_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><b>case </b></a><a NAME="E1:62_1_1_2"/><i><font color="Green">E66</font></i>: 
(  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E2:62_1_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E2:62_1_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan3.html#E5:62_1_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:62_1_1_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E10:62"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan3.html#E4:62"><i><font color="Green">E53</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:62_1_1_2_1"/><i><font color="Green">E70</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>and </b><br/><a NAME="E6:62_1_1_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/><i><font color="Green">E72</font></i>: <font color="Maroon">p</font> = 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><a NAME="E8:62_1_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E10:62"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan3.html#E15:62"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan3.html#E16:62"><i><font color="Green">E77</font></i></a>, , <a class="txt" href="jordan3.html#E4:62"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E9:62_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan3.html#E16:62"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E10:62_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan3.html#E2:62_1_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan3.html#E5:62_1_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan3.html#E13:62"><i><font color="Green">E75</font></i></a>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div><b>assume </b><a NAME="E3:62_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:62_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan3.html#E11:62"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jordan3.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:62"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> <a href="jordan3.html#R1" target="_self">is_S-Seq_joining</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:62"/><i><font color="Green">E74</font></i>: 
 <a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 
<b>by </b><i>, , <a class="txt" href="jordan3.html#E7:62"><i><font color="Green">E62</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_2_1"/>
(  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> or (  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_1"><b>case </b></a><a NAME="E1:62_2_1_1"/>
 <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:62_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:62_2_1_1"/>
 ex <font color="Olive">i1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i1</font> &amp; <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>,<font color="Olive">i1</font> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_2"><b>case </b></a><a NAME="E1:62_2_1_2"/>
(  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:62_2_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E3:62_2_1_2"/>
 ex <font color="Olive">i1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i1</font> &amp; <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>,<font color="Olive">i1</font> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:62"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/>
<a NAME="E14:62"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> &amp;  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E15:62"/><b>then </b><i><font color="Green">E76</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:62"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E10:62"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E17:62"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" target="_self">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E15:62"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E18:62"/>
 <a href="jordan3.html#K5" target="_self">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="jordan3.html#R1" target="_self">is_S-Seq_joining</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i>, , <a class="txt" href="jordan3.html#E11:62"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan3.html#E13:62"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan3.html#E16:62"><i><font color="Green">E77</font></i></a>, </i>;<br/>


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