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

<b>let </b><font color="Maroon">f</font> be  non <a href="xboole_0.html#V1">empty</a> <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="E1:65"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">f</font> is <a href="jordan23.html#V1" target="_self">almost-one-to-one</a> &amp; <font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Maroon">f</font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">f</font> is <a href="topreal1.html#V3">s.n.c.</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="E2:65"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:65"/><i><font color="Green">E46</font></i>: 
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> or (  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">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">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">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="E4:65"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="jordan3.html#K5">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">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">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="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan3.html#D8">JORDAN3:def 8</a></i>;<br/>
<a NAME="E5:65"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <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="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E6:65"/>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <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="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E7:65"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">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="E8:65"/><i><font color="Green">E50</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">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="E9:65"/><i><font color="Green">E51</font></i>: 
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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:65"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> is  <a href="jordan23.html#V1" target="_self">almost-one-to-one</a> <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i><a class="ref" href="jordan23.html#T3" target="_self">Th3</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1"/>
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> or (  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">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">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">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="65_1_1_1"><b>case </b></a><a NAME="E1:65_1_1_1"/>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:65_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">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">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="E3:65_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="E4:65_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">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan23.html#D3" target="_self">Def3</a>, <a class="txt" href="jordan23.html#E3:65"><i><font color="Green">E46</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:65_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2"><b>case </b></a><a NAME="E1:65_1_1_2"/><i><font color="Green">E62</font></i>: 
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">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">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">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">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1_1_2_1"/><i><font color="Green">E64</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">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="E2:65_1_1_2_1"/><b>then </b><i><font color="Green">E65</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">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="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:65_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">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">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="jordan23.html#E3:65"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#D6">JORDAN3:def 6</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="E5:65_1_1_2_1"/><i><font color="Green">E66</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="E6:65_1_1_2_1"/><i><font color="Green">E67</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">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">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">E68</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">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="E8:65_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="jordan23.html#E3:65"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan23.html#E7:65"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E8:65"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan23.html#E9:65"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan3.html#D6">JORDAN3:def 6</a></i>;<br/><a NAME="E9:65_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="jordan23.html#E8:65"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E10:65_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, , <a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="txt" href="jordan23.html#E5:65"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E9:65"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan23.html#E10:65"><i><font color="Green">E52</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="E3:65_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="E4:65_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan23.html#D3" target="_self">Def3</a>, <a class="txt" href="jordan23.html#E4:65"><i><font color="Green">E47</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E12:65"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> <a href="jordan3.html#R1">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><a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, , </i>;<br/>
<a NAME="E13:65"/><i><font color="Green">E70</font></i>: 
 <a href="jordan3.html#K3">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="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan23.html#E2:65"><i><font color="Green">E45</font></i></a>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_2_1"/>
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> or (  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">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">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">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="65_2_1_1"><b>case </b></a><a NAME="E1:65_2_1_1"/>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:65_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">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan3.html#T64">JORDAN3:64</a></i>;<br/><b>hence </b><a NAME="E3:65_2_1_1"/>
 ex <font color="Olive">i1</font> being   <a href="subset_1.html#M1">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">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">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="65_2_1_2"><b>case </b></a><a NAME="E1:65_2_1_2"/>
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">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">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">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="E2:65_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">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan3.html#T66">JORDAN3:66</a></i>;<br/><b>hence </b><a NAME="E3:65_2_1_2"/>
 ex <font color="Olive">i1</font> being   <a href="subset_1.html#M1">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">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">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="E15:65"/><b>then </b><i><font color="Green">E71</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">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="E16:65"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> <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">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#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E17:65"/><b>then </b><i><font color="Green">E72</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">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="E18:65"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K3">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="jordan23.html#E3:65"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#D3">JORDAN3:def 3</a></i>;<br/>
<a NAME="E19:65"/><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">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="jordan23.html#E7:65"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E20:65"/>
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="jordan3.html#R1">is_S-Seq_joining</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E4:65"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E5:65"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E8:65"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan3.html#T67">JORDAN3:67</a></i>;<br/>


</div>
