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

<b>let </b><font color="Maroon">f</font> be  <a href="finseq_6.html#V1">circular</a> <a href="goboard5.html#V1">s.c.c.</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:37"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:37"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E6:37"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:37"/><i><font color="Green">E55</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:37"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E9:37"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:37"/><b>then </b>
<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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:37"/><b>then </b><i><font color="Green">E62</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#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E12:37"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:37"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1"/>
( ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 ) or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1 or ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) or ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) or ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) )
 </span><b>by </b><i><a class="ref" href="goboard7.html#T1" target="_self">Th1</a>, <a class="txt" href="goboard7.html#E1:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:37_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:37_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E3:37_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:37_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E5:37_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:37_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:37_1_1"/><i><font color="Green">E74</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="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E8:37_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="ref" href="goboard7.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E9:37_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:37_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard7.html#T6" target="_self">Th6</a>, <a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/><a NAME="E11:37_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E12:37_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/><a NAME="E13:37_1_1"/><b>then </b>
<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#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="txt" href="goboard7.html#E2:37"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/><b>hence </b><a NAME="E14:37_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:37_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:37_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E3:37_1_2"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:37_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard7.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E5:37_1_2"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="ref" href="goboard7.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E6:37_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/><a NAME="E7:37_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E8:37_1_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#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="txt" href="goboard7.html#E2:37"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/><b>hence </b><a NAME="E9:37_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_3"><b>suppose </b></a><a NAME="E1:37_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:37_1_3"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E3:37_1_3"/><b>then </b><i><font color="Green">E84</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:37_1_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/><a NAME="E5:37_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E6:37_1_3"/>
<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#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/><b>hence </b><a NAME="E7:37_1_3"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="goboard7.html#E2:37"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:37_1_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:37_1_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E3:37_1_4"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:37_1_4"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/><a NAME="E5:37_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E6:37_1_4"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:37_1_5"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:37_1_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E3:37_1_5"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:37_1_5"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/><a NAME="E5:37_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E6:37_1_5"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_6"><b>suppose </b></a><b>that </b><a NAME="E1:37_1_6"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:37_1_6"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E3:37_1_6"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E4:37_1_6"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E5:37_1_6"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:37_1_6"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="txt" href="goboard7.html#E2:37"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E7:37_1_6"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard7.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E8:37_1_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard7.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:37_1_6"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="goboard7.html#E2:37"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/><a NAME="E10:37_1_6"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:37_1_6"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,1 <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="txt" href="goboard7.html#E2:37"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/><a NAME="E12:37_1_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E13:37_1_6"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard7.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
