<?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><b>that </b><br/><a NAME="E1:57"/><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> )
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green">E46</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
 <b>and </b><br/><a NAME="E4:57"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<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> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:57"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <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> <font color="Maroon">f</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/>
<a NAME="E7:57"/><i><font color="Green">E49</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E8:57"/><i><font color="Green">E50</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="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E9:57"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E10:57"/><b>then </b><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="E11:57"/><b>then </b><i><font color="Green">E52</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>,1,<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> <a href="hidden.html#R1">=</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="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E12:57"/><b>then </b><i><font color="Green">E53</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>,1,<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> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E13:57"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E14:57"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:57"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<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> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan23.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan3.html#T17">JORDAN3:17</a></i>;<br/>
<a NAME="E16:57"/><b>then </b>
<font color="Maroon">g</font> <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> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E17:57"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<i><font color="Green">E63</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> = 
<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>,1,<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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<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>,1,<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="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E19:57"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">g</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">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E20:57"/><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> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan23.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E21:57"/><i><font color="Green">E66</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/>;<br/>
<a NAME="E22:57"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">x1</font>, <font color="Maroon">n2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:57_2"/><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <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="txt" href="jordan23.html#E4:57"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>now </b></a><div class="add"><i><font color="Green">E70</font></i>: <font color="Maroon">g</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">g</font></span>)</span> = 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_2_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or ( <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_2_1_1"><b>case </b></a><a NAME="E1:57_2_1_2_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_2_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:57_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n2</font> <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="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:57_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">n2</font> <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>,1,<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>
 <b>by </b><i><a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:57_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:57_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</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"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, , <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E6:57_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E74</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> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E4:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E7:57_2_1_2_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">n2</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="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:57_2_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="jordan23.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:57_2_1_2_1_1_1"/><b>then </b>
<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="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E2:57"><i><font color="Green">E45</font></i></a>, , <a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="jordan23.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E10:57_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:57_2_1_2_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_2_1_2"><b>case </b></a><a NAME="E1:57_2_1_2_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_2_1_2_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:57_2_1_2_1_2_1"/><b>then </b>
<font color="Maroon">n1</font> <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="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:57_2_1_2_1_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">n1</font> <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>,1,<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>
 <b>by </b><i><a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:57_2_1_2_1_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:57_2_1_2_1_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</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"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, , <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E6:57_2_1_2_1_2_1"/><b>then </b><i><font color="Green">E79</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> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E4:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E7:57_2_1_2_1_2_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">n1</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="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:57_2_1_2_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="jordan23.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:57_2_1_2_1_2_1"/><b>then </b>
<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="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E2:57"><i><font color="Green">E45</font></i></a>, , <a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="jordan23.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E10:57_2_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E1:57"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:57_2_1_2_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_2_1_3"><b>case </b></a><b>that </b><a NAME="E1:57_2_1_2_1_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E2:57_2_1_2_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E3:57_2_1_2_1_3"/>
<font color="Maroon">n1</font> <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="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:57_2_1_2_1_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">n1</font> <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>,1,<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>
 <b>by </b><i><a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:57_2_1_2_1_3"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">n1</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="jordan23.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E85</font></i>: <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
<b>by </b><i>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</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#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan3.html#T32">JORDAN3:32</a></i>
;<br/><a NAME="E7:57_2_1_2_1_3"/>
<font color="Maroon">n2</font> <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="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E8:57_2_1_2_1_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">n2</font> <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>,1,<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>
 <b>by </b><i><a class="ref" href="jordan23.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:57_2_1_2_1_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">n2</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="jordan23.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E88</font></i>: <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
<b>by </b><i>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan23.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</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#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan23.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan3.html#T32">JORDAN3:32</a></i>
;<br/><a NAME="E11:57_2_1_2_1_3"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:57_2_1_2_1_3"/><b>then </b>
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E13:57_2_1_2_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</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#E4:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan23.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan23.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan23.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:57_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:57"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57_3"/><i><font color="Green">E90</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E2:57_3"/><b>then </b>
<font color="Maroon">j</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> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T10">JORDAN3:10</a></i>;<br/>
<a NAME="E3:57_3"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan3.html#T51">JORDAN3:51</a></i>;<br/>
<a NAME="E4:57_3"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</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">g</font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E5:57_3"/><i><font color="Green">E93</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan3.html#T11">JORDAN3:11</a></i>;<br/>
<a NAME="E6:57_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan3.html#T51">JORDAN3:51</a></i>;<br/>
<a NAME="E7:57_3"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E8:57_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">j</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> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan3.html#T10">JORDAN3:10</a></i>;<br/>
<a NAME="E9:57_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E10:57_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E11:57_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:57_3"/><i><font color="Green">E98</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="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E13:57_3"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E14:57_3"/><b>then </b>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E15:57_3"/><b>then </b>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan23.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:57_3"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1"><b>now </b></a><div class="add"><a NAME="E1:57_3_1"/><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</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>,1,<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="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E2:57_3_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="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/><a NAME="E3:57_3_1"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <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="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E14:57"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_3_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <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="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="jordan23.html#T9" target="_self">Th9</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_1_1_1"><b>case </b></a><a NAME="E1:57_3_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:57_3_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E14:57"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_3_1_1_1_1"/>
contradiction
 <b>by </b><i>, <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="57_3_1_1_1_2"><b>case </b></a><a NAME="E1:57_3_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <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="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:57_3_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:57_3_1_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 2 <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="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:57_3_1_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">j</font> <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">p</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan23.html#E14:57"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:57_3_1_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">j</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> <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="E6:57_3_1_1_1_2"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan23.html#T10" target="_self">Th10</a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E7:57_3_1_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan23.html#E17:57"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan23.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan23.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan23.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan3.html#T32">JORDAN3:32</a></i>
;<br/><b>hence </b><a NAME="E9:57_3_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="ref" href="jordan23.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan23.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan23.html#E18:57"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan23.html#E19:57"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:57_3_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:57_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E24:57"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">j1</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">j2</font> holds <br/> <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">j1</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">j2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j2</font>
 ;<br/>

<a NAME="E2:57_4"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:57_4"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_4_1_1"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> 0 or ( ( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 ) &amp; <font color="Maroon">j2</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> <font color="Maroon">g</font> ) or <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 </span><b>by </b><i><a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_4_1_1_1"><b>case </b></a><a NAME="E1:57_4_1_1_1"/>
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:57_4_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:57_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:57_4_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_4_1_1_2"><b>case </b></a><b>that </b><a NAME="E1:57_4_1_1_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 <b>and </b><br/><a NAME="E2:57_4_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">j2</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> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E3:57_4_1_1_2"/>
<font color="Maroon">j2</font> <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="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:57_4_1_1_2"/><b>then </b>
<font color="Maroon">j1</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> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:57_4_1_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j1</font>
 <b>by </b><i>, , <a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan3.html#T51">JORDAN3:51</a></i>;<br/><a NAME="E6:57_4_1_1_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:57_4_1_1_2"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:57_4_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:57_4_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j2</font>
 <b>by </b><i>, , <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan3.html#T51">JORDAN3:51</a></i>;<br/><a NAME="E10:57_4_1_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E11:57_4_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j2</font>
 <b>by </b><i>, <a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E12:57_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E13:57_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="jordan23.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E14:57_4_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_4_1_1_3"><b>case </b></a><a NAME="E1:57_4_1_1_3"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:57_4_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:57_4_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:57_4_1_1_3"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57_4"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E25:57"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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> <font color="Maroon">g</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57_5"/><i><font color="Green">E143</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</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> <font color="Maroon">g</font> )
 ;<br/>

<div><i><font color="Green">E144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5_1"><b>now </b></a><div class="add"><a NAME="E1:57_5_1"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 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="jordan23.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E2:57_5_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:57_5_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="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/><a NAME="E4:57_5_1"/><b>then </b>
<font color="Maroon">j</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="txt" href="jordan23.html#E10:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:57_5_1"/>
<font color="Maroon">j</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> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:57_5"/><i><font color="Green">E146</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan3.html#T51">JORDAN3:51</a></i>;<br/>
<a NAME="E4:57_5"/><i><font color="Green">E147</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E5:57_5"/><i><font color="Green">E148</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</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> <font color="Maroon">j</font></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"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E7:57"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E6:57_5"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</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"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</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"><font color="Maroon">j</font> <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#E7:57"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E8:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:57_5"/>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <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#E10:57"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan23.html#E11:57"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan23.html#E12:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="jordan3.html#T36">JORDAN3:36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:57"/><b>then </b>
( <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 &amp; <font color="Maroon">g</font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">g</font> is <a href="topreal1.html#V3">s.n.c.</a> &amp; <font color="Maroon">g</font> is <a href="topreal1.html#V1">special</a> )
 
<b>by </b><i>, <a class="txt" href="jordan23.html#E3:57"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan23.html#E4:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E6:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/>
<a NAME="E27:57"/><b>then </b>
<font color="Maroon">g</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E28:57"/>
<font color="Maroon">g</font> <a href="jordan3.html#R1">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="jordan23.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan3.html#D3">JORDAN3:def 3</a></i>;<br/>


</div>
