<?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">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:20"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <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#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E5:20"/>
<font color="Maroon">i</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#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:20"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</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#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:20"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:20"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:20"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:20"/><b>then </b><i><font color="Green">E44</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> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E11:20"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E13:20"/><b>then </b>
<font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E14:20"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 
;<br/>
<a NAME="E15:20"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E16:20"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">i1</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">i1</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font>

<br/>.= 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="jordan4.html#T2" target="_self">Th2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<b>thus </b> <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="sppol_2.html#T2">SPPOL_2:2</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><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> <span class="p5">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i2</font>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan4.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan4.html#T3" target="_self">Th3</a>, </i>
;<br/>


</div>
