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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM8">FinSequence</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="E1:51"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V1">special</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></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="E2:51"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:51"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T39">SPPOL_2:39</a></i>;<br/>
<a NAME="E4:51"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:51"><i><font color="Green">E27</font></i></a>, <a class="ref" href="topreal1.html#T28">TOPREAL1:28</a></i>;<br/>
<a NAME="E5:51"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E6:51"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:51"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:51"/><i><font color="Green">E31</font></i>: 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:51"><i><font color="Green">E26</font></i></a>, <a class="ref" href="jordan23.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E9:51"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E10:51"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>
<a NAME="E11:51"/><i><font color="Green">E34</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:51"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E12:51"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:51"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E6:51"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan23.html#E7:51"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:51"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T12">SPRECT_2:12</a></i>;<br/>
<a NAME="E15:51"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></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#E2:51"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan5b.html#T32">JORDAN5B:32</a></i>;<br/>
<a NAME="E16:51"/><i><font color="Green">E38</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></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#E12:51"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan23.html#E11:51"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1"/>
(  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="sppol_1.html#V2">vertical</a> or  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="sppol_1.html#V1">horizontal</a> )
 </span><b>by </b><i><a class="txt" href="jordan23.html#E1:51"><i><font color="Green">E26</font></i></a>, <a class="ref" href="sppol_1.html#T41">SPPOL_1:41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><b>case </b></a><a NAME="E1:51_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E15:51"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan23.html#E16:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sppol_1.html#T64">SPPOL_1:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2"><b>case </b></a><a NAME="E1:51_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="sppol_1.html#V1">horizontal</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E15:51"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan23.html#E16:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sppol_1.html#T63">SPPOL_1:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1"><b>suppose </b></a><a NAME="E1:51_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:51_2_1"/><b>then </b>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/><b>hence </b><a NAME="E3:51_2_1"/>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan23.html#E3:51"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan23.html#E8:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan23.html#E17:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E9:51"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan23.html#E10:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan23.html#E14:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="goboard2.html#T13">GOBOARD2:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_2"><b>suppose </b></a><a NAME="E1:51_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_2_2"/>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan23.html#E8:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
