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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<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">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:10"/><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#R1">&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">f1</font> )
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green">E33</font></i>: 
1 <a href="xxreal_0.html#R1">&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/>
<a NAME="E3:10"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:10"/><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">f1</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</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>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E5:10"/><b>then </b>
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">f1</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>then </b><i><font color="Green">E35</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</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#K1">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span></span>)</span> = 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span></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="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <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="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="jordan3.html#T27">JORDAN3:27</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> = 
<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#K3">+</a> 1</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="nat_1.html#K1">+</a> 1

;<br/>
<b>hence </b><a NAME="E8:10"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</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#K1">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>


</div>
