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

<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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">p1</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>, <font color="Maroon">p2</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>, <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>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">P</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="topreal4.html#T3">TOPREAL4:3</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <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><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E32</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:32"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>hence </b><a NAME="E2:32_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:32"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> &amp; 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> )
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E11:32"/><i><font color="Green">E43</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:32"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</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="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:32"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <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="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:32"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:32"/><i><font color="Green">E46</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="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:32"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/><font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>;<br/>
<a NAME="E17:32"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E18:32"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:32"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>

;<br/>
<a NAME="E21:32"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E22:32"/><b>then </b><i><font color="Green">E76</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<a NAME="E23:32"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="jordan20.html#E11:32"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E25:32"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 ;<br/>
<a NAME="E26:32"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">k</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="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E27:32"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E28:32"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E29:32"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E30:32"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E31:32"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:32"/><i><font color="Green">E85</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="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E16:32"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:32"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E34:32"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</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="jordan20.html#E15:32"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E35:32"/><b>then </b><i><font color="Green">E86</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="nat_1.html#K1">+</a> <font color="Maroon">k</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/>
<a NAME="E36:32"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">pk</font> = <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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> as   <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/>
<a NAME="E38:32"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">pk</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">kn</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1"/>
( <font color="Maroon">pk</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> or <font color="Maroon">pk</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> )
 </span><b>by </b><i><a class="ref" href="jordan20.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1"><b>case </b></a><a NAME="E1:32_3_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">pk</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">p44</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_1_1_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">pk</font> <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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1"/><i><font color="Green">E94</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_1"/>
for <font color="Olive">p5</font> being  <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>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_1_1"/><i><font color="Green">E96</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p8</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> or <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, , <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_1_1_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:32_3_1_1_1_1_1"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , , <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">p44</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><div><i><font color="Green">E103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_1"/>
<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_1"/><b>then </b>
for <font color="Olive">p9</font> being  <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>  st <font color="Olive">p9</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> holds <br/><font color="Olive">p9</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E3:32_3_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_1_1_1_2"/><i><font color="Green">E105</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> as   <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/><a NAME="E6:32_3_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">p5</font> being  <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>  st  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_2"/><i><font color="Green">E111</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_2"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p8</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:32_3_1_1_1_1_2_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_1_2_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_2_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> or <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_2_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_2_2_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, , <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_2_2_1_1_2"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:32_3_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_1_1_1_2_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">g</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<font color="Maroon">i</font> as   <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/><a NAME="E10:32_3_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="txt" href="jordan20.html#E15:32"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3"/>
 ex <font color="Olive">p51</font> being  <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> st <br/>(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not <font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> )
 ;<br/><b>then consider </b><font color="Maroon">p51</font> being   <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><b> such that </b><br/><a NAME="E3:32_3_1_1_1_1_2_3"/><i><font color="Green">E120</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> )
 ;<br/><a NAME="E4:32_3_1_1_1_1_2_3"/>
<font color="Maroon">p51</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">Y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:32_3_1_1_1_1_2_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">p51</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y3</font> &amp; <font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i5</font></span>)</span> where <font color="Olive">i5</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i5</font> &amp; <font color="Olive">i5</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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:32_3_1_1_1_1_2_3"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">Y3</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">kk</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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/>;<br/><a NAME="E9:32_3_1_1_1_1_2_3"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_2_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">kk</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="E11:32_3_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">kk</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_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2_3"/><i><font color="Green">E127</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3"/><i><font color="Green">E129</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">kk</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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3"/><i><font color="Green">E130</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_2_3"/><i><font color="Green">E131</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">p51</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T25">JORDAN5C:25</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E18:32_3_1_1_1_1_2_3"/>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:32_3_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E20:32_3_1_1_1_1_2_3"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_3_1_1"/>
( <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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> or ( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> &amp; <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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> ) )
 </span><b>by </b><i>, , , <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E135</font></i>: 
<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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">k2</font> = <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E136</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 <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="E3:32_3_1_1_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_2_3_1_1_1_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_1_1_2_3_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E138</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <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/><i><font color="Green">E139</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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E26:32"><i><font color="Green">E79</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1

;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>]
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E140</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E17:32"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E141</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E143</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="txt" href="jordan20.html#E21:32"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2_3_1_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Olive">p8</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, </i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E146</font></i>: 
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_1_3"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_3_1_1_1_3"/>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i7</font></span>)</span> where <font color="Olive">i7</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i7</font> &amp; <font color="Olive">i7</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> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_1_2_3_1_1_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E18:32_3_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E20:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E21:32_3_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E23:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E24:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E154</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</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">kk</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">kk</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="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><a NAME="E25:32_3_1_1_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , <a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, , </i>;<br/><a NAME="E26:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , , <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan20.html#E31:32"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E28:32_3_1_1_1_1_2_3_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E156</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> &amp; <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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">k2</font> = <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E157</font></i>: 
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_1"/>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i7</font></span>)</span> where <font color="Olive">i7</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i7</font> &amp; <font color="Olive">i7</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> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E26:32"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E159</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Olive">p8</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E160</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_2_3_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E22:32"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E161</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</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="E4:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E164</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_2_3_1_1_2_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E26:32"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, </i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span>,<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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E166</font></i>: 
not  <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#R1">misses</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E169</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">kk</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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E170</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V3">s.n.c.</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<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> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E174</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T2">JORDAN2C:2</a></i>;<br/><div><i><font color="Green">E175</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E177</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_1_1_1_2_3_1_1_2_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><i><font color="Green">E178</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1 )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">f</font>,<font color="Maroon">kk</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">f</font>,<font color="Maroon">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E21:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E22:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</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">kk</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">kk</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="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E23:32_3_1_1_1_1_2_3_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , , , , , , </i>;<br/></div><b>end;</b></div><a NAME="E11:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E179</font></i>: 
<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<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#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E31:32"><i><font color="Green">E83</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">kk</font>

;<br/><a NAME="E13:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="txt" href="jordan20.html#E21:32"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_1_1_1_2_3_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , , <a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E85</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:32_3_1_1_1_1_2_3"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E15:32_3_1_1_1_1_2"/>
for <font color="Olive">p5</font> being  <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>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_4"/><i><font color="Green">E181</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_4"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_4_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 </span><b>by </b><i>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_4_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_4_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_4_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_4_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_1_2_4"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:32_3_1_1_1_1_2"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:32_3_1_1"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2"><b>case </b></a><a NAME="E1:32_3_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">pk</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">p44</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_2_1_1_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">pk</font> <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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_1"/><i><font color="Green">E186</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_1"/>
for <font color="Olive">p5</font> being  <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>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_1_1"/><i><font color="Green">E188</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p8</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:32_3_1_2_1_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_2_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E190</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> or <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, , <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_2_1_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:32_3_1_2_1_1_1"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , , <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">p44</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><div><i><font color="Green">E192</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_1"/>
<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_1"/><b>then </b>
for <font color="Olive">p9</font> being  <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>  st <font color="Olive">p9</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> holds <br/><font color="Olive">p9</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E3:32_3_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_2_1_1_2"/><i><font color="Green">E193</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> as   <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/><a NAME="E6:32_3_1_2_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E261</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2"/><i><font color="Green">E262</font></i>: 
for <font color="Olive">p5</font> being  <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>  st  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_2"/><i><font color="Green">E263</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_2"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p8</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E264</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:32_3_1_2_1_1_2_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_2_1_1_2_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E265</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> or <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_2_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_2_2_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, , <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_2_2_1_1_2"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p</font>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:32_3_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_2_1_1_2_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">g</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<font color="Maroon">i</font> as   <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/><a NAME="E10:32_3_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E11:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E266</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="txt" href="jordan20.html#E15:32"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E267</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_2_1_1_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><i><font color="Green">E269</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3"/>
 ex <font color="Olive">p51</font> being  <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> st <br/>(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not <font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> )
 ;<br/><b>then consider </b><font color="Maroon">p51</font> being   <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><b> such that </b><br/><a NAME="E3:32_3_1_2_1_1_2_3"/><i><font color="Green">E270</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 ;<br/><a NAME="E4:32_3_1_2_1_1_2_3"/>
<font color="Maroon">p51</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">Y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:32_3_1_2_1_1_2_3"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">p51</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y3</font> &amp; <font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i5</font></span>)</span> where <font color="Olive">i5</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i5</font> &amp; <font color="Olive">i5</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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:32_3_1_2_1_1_2_3"/><i><font color="Green">E272</font></i>: 
( <font color="Maroon">Y3</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">kk</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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/>;<br/><a NAME="E9:32_3_1_2_1_1_2_3"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E10:32_3_1_2_1_1_2_3"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">kk</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="E11:32_3_1_2_1_1_2_3"/><b>then </b>
<font color="Maroon">kk</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="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E275</font></i>: 
<font color="Maroon">kk</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/><a NAME="E13:32_3_1_2_1_1_2_3"/><i><font color="Green">E276</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E277</font></i>: 
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3"/><i><font color="Green">E278</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">kk</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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E16:32_3_1_2_1_1_2_3"/><i><font color="Green">E279</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3"/><i><font color="Green">E280</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">p51</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T25">JORDAN5C:25</a></i>;<br/><a NAME="E18:32_3_1_2_1_1_2_3"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E19:32_3_1_2_1_1_2_3"/>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E20:32_3_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E282</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E21:32_3_1_2_1_1_2_3"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_3_1_1"/>
( <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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> or ( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> ) )
 </span><b>by </b><i>, , , <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E284</font></i>: 
<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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">k2</font> = <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E285</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E286</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_1_1"/>
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 <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="E3:32_3_1_2_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_2_3_1_1_1_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_2_1_1_2_3_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E287</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <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/><i><font color="Green">E288</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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E26:32"><i><font color="Green">E79</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1

;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>]
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E289</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E17:32"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E10:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E290</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E304</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="txt" href="jordan20.html#E21:32"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E305</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_2_1_1_2_3_1_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Olive">p8</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E306</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, </i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E307</font></i>: 
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_1_3"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_3_1_1_1_3"/>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i7</font></span>)</span> where <font color="Olive">i7</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i7</font> &amp; <font color="Olive">i7</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> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_2_1_1_2_3_1_1_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E309</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E310</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E18:32_3_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E311</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E20:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E312</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E21:32_3_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E23:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E313</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E24:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E314</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</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">kk</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">kk</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="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><a NAME="E25:32_3_1_2_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , <a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, , </i>;<br/><a NAME="E26:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , , <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan20.html#E31:32"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E28:32_3_1_2_1_1_2_3_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E315</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">k2</font> = <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E316</font></i>: 
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_1"/><i><font color="Green">E317</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_1"/>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i7</font></span>)</span> where <font color="Olive">i7</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i7</font> &amp; <font color="Olive">i7</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> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E26:32"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E318</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Olive">p8</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E319</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
 <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E22:32"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E320</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E321</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E322</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</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="E4:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E323</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_2_3_1_1_2_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E26:32"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E81</font></i></a>, </i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font></span>)</span>,<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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E325</font></i>: 
not  <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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#R1">misses</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E11:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E326</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">kk</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">kk</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E327</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</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">i</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E13:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E328</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V3">s.n.c.</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E329</font></i>: 
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E330</font></i>: 
<font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E16:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E38:32"><i><font color="Green">E88</font></i></a>, , <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<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> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E331</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T2">JORDAN2C:2</a></i>;<br/><div><i><font color="Green">E332</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E333</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E334</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_2_1_1_2_3_1_1_2_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><i><font color="Green">E335</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1 )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">f</font>,<font color="Maroon">kk</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">f</font>,<font color="Maroon">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E21:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E22:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</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">kk</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">kk</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">kk</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="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E23:32_3_1_2_1_1_2_3_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , , , , , , </i>;<br/></div><b>end;</b></div><a NAME="E11:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E336</font></i>: 
<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<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#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E31:32"><i><font color="Green">E83</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">kk</font>

;<br/><a NAME="E13:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">kk</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E16:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E337</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">kk</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , <a class="txt" href="jordan20.html#E21:32"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_2_1_1_2_3_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, , , , <a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E85</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E23:32_3_1_2_1_1_2_3"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E15:32_3_1_2_1_1_2"/>
for <font color="Olive">p5</font> being  <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>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_4"/><i><font color="Green">E338</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">pk</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_4"/><b>then </b><i><font color="Green">E339</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_4_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 </span><b>by </b><i>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_4_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_4_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_4_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_4_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_2_1_1_2_4"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:32_3_1_2_1_1_2"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E28</font></i></a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:32_3_1_2"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E41:32"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 ;<br/>


</div>
