<?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>, <font color="Maroon">g</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>;<br/>



<b>assume </b><a NAME="E1:84"/><i><font color="Green">E50</font></i>: 
( <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> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">g</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 )
 ;<br/>

<a NAME="E2:84"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E3:84"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E4:84"/><i><font color="Green">E53</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="E5:84"/><i><font color="Green">E54</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:84"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:84"/><b>then </b><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E8:84"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T23" target="_self">Th23</a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E9:84"/><i><font color="Green">E64</font></i>: 
 <a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</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#E8:84"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E10:84"/><b>then </b><i><font color="Green">E65</font></i>: 
1 <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#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E11:84"/>
 <a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="jordan3.html#R1" target="_self">is_S-Seq_joining</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:84"/><b>then </b><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</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">g</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="jordan3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E13:84"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</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">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>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E14:84"/><i><font color="Green">E68</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#K4" target="_self">R_Cut</a> <font color="Maroon">g</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="E15:84"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E16:84"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="txt" href="jordan3.html#E2:84"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E17:84"/>
<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> <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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</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"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T26" target="_self">Th26</a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E18:84"/><b>then </b>
<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> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan3.html#T26" target="_self">Th26</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E19:84"/><b>then </b>
<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> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E20:84"/><b>then </b><i><font color="Green">E70</font></i>: 
<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> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/>
<a NAME="E21:84"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E22:84"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E23:84"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span>,1
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E24:84"/><b>then </b>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</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="E25:84"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E26:84"/><b>then </b>
<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> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E27:84"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E28:84"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="jordan3.html#E14:84"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E29:84"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E30:84"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E31:84"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font></span>)</span> <a href="jordan3.html#R1" target="_self">is_S-Seq_joining</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T27" target="_self">Th27</a>, , , </i>;<br/>


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