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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E3:81"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:81"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:81"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E6:81"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:81"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:81"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E9:81"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:81_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/><b>hence </b><a NAME="E3:81_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sppol_2.html#E7:81"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="sppol_2.html#T50" target="_self">Th50</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E12:81"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:81"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E15:81"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<a NAME="E16:81"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font> as  <a href="funct_1.html#V2">one-to-one</a> <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T59">FINSEQ_5:59</a></i>;<br/>
<a NAME="E18:81"/><i><font color="Green">E77</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><a NAME="E2:81_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/><a NAME="E3:81_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>;<br/><a NAME="E4:81_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:81_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:81_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E8:81_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E2:81_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_5.html#T55">FINSEQ_5:55</a></i>;<br/><a NAME="E9:81_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E2:81_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_5.html#T54">FINSEQ_5:54</a></i>;<br/><a NAME="E10:81_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sppol_2.html#E18:81"><i><font color="Green">E77</font></i></a>, <a class="txt" href="sppol_2.html#E4:81_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="sppol_2.html#E8:81_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/><a NAME="E11:81_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E12:81_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:81"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:81"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/>
<a NAME="E21:81"/><b>then </b><i><font color="Green">E78</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:81"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E4:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E9:81"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_5.html#T44">FINSEQ_5:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E2:81_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T43">FINSEQ_5:43</a></i>;<br/><b>hence </b><a NAME="E3:81_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:81"><i><font color="Green">E75</font></i></a>, <a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E22:81"><i><font color="Green">E79</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E24:81"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E25:81"/><b>then </b><i><font color="Green">E80</font></i>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E21:81"><i><font color="Green">E78</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E26:81"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/>
<a NAME="E27:81"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T21">FINSEQ_5:21</a>, <a class="ref" href="finseq_5.html#T58">FINSEQ_5:58</a></i>;<br/>
<a NAME="E28:81"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E29:81"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E30:81"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E31:81"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E32:81"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E5:81"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E28:81"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E33:81"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="sppol_2.html#E25:81"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/>
<a NAME="E34:81"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></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="E35:81"/>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>13</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
;<br/>
<a NAME="E36:81"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E34:81"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E37:81"/><i><font color="Green">E84</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E38:81"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E39:81"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E40:81"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E41:81"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E42:81"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E43:81"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E44:81"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="sppol_2.html#E15:81"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E45:81"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E40:81"><i><font color="Green">E85</font></i></a>, <a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E46:81"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E47:81"/><i><font color="Green">E90</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E48:81"/><i><font color="Green">E91</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E44:81"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E49:81"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font> is <a href="topreal1.html#V2">unfolded</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="sppol_2.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E50:81"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E51:81"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font>
 
;<br/>
<a NAME="E52:81"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E53:81"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<i><font color="Green">E96</font></i>: <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T57">FINSEQ_5:57</a></i>
;<br/>
<a NAME="E55:81"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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">c<sub>6</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E56:81"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E51:81"><i><font color="Green">E93</font></i></a>, <a class="txt" href="sppol_2.html#E52:81"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:81_5"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self">Th22</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:81_5"/><i><font color="Green">E100</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E5:81_5"/>
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:81_5"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E7:81_5"/><i><font color="Green">E102</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E8:81_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E9:81_5"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,1
 <b>by </b><i><a class="txt" href="sppol_2.html#E7:81_5"><i><font color="Green">E102</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E10:81_5"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_5"><i><font color="Green">E100</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:81_5"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E13:81_5"/><i><font color="Green">E105</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E12:81_5"><i><font color="Green">E104</font></i></a>, <a class="ref" href="sppol_2.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E14:81_5"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:81_5"><i><font color="Green">E101</font></i></a>, <a class="txt" href="sppol_2.html#E9:81_5"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E15:81_5"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,1 <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><a NAME="E16:81_5"/><i><font color="Green">E107</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:81"><i><font color="Green">E74</font></i></a>, <a class="txt" href="sppol_2.html#E37:81"><i><font color="Green">E84</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_5_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>15</sub></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="81_5_1_1"><b>suppose </b></a><a NAME="E1:81_5_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:81_5_1_1"/><i><font color="Green">E109</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81_5"><i><font color="Green">E107</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_5_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_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="E1:81_5_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 )
 </span><b>by </b><i><a class="txt" href="sppol_2.html#E2:81_5_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_5_1_1_1_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:81_5_1_1_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_5_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_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="E1:81_5_1_1_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 )
 </span><b>by </b><i><a class="txt" href="sppol_2.html#E1:81_5_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="sppol_2.html#E2:81_5_1_1_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_5_1_1_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:81_5_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_5_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="finseq_5.html#T53">FINSEQ_5:53</a></i>;<br/><b>hence </b><a NAME="E3:81_5_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_5"><i><font color="Green">E99</font></i></a>, <a class="ref" href="topreal1.html#T28">TOPREAL1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_5_1_1_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:81_5_1_1_1_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:81_5_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font>,<span class="p2">(<span class="default">1 <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">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default">1 <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="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:81_5_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:81_5"><i><font color="Green">E101</font></i></a>, <a class="txt" href="sppol_2.html#E9:81_5"><i><font color="Green">E103</font></i></a>, <a class="txt" href="sppol_2.html#E13:81_5"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_5_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_5_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:81_5_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 2
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:81_5_1_1_1_1_1_1_1_2"/>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_5_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/><b>hence </b><a NAME="E7:81_5_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sppol_2.html#E18:81"><i><font color="Green">E77</font></i></a>, <a class="txt" href="sppol_2.html#E5:81_5_1_1_1_1_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:81_5_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_5_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:81_5_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_5_1_1"><i><font color="Green">E108</font></i></a></i>;<br/><b>hence </b><a NAME="E3:81_5_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:81_5"><i><font color="Green">E106</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:81_5_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_5_1_2"><b>suppose </b></a><a NAME="E1:81_5_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:81_5_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:81_5_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81_5"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:81_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:81_5"><i><font color="Green">E106</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E58:81"/><i><font color="Green">E99</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E51:81"><i><font color="Green">E93</font></i></a>, <a class="txt" href="sppol_2.html#E53:81"><i><font color="Green">E95</font></i></a>, <a class="txt" href="sppol_2.html#E54:81"><i><font color="Green">E96</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_6"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:81_6"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E40:81"><i><font color="Green">E85</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:81_6"/><b>then </b><i><font color="Green">E101</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:81_6_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:81_6_1_1"/><i><font color="Green">E103</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E58:81"><i><font color="Green">E99</font></i></a>, <a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_6"><i><font color="Green">E101</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:81_6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b> <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1 = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="sppol_2.html#T10" target="_self">Th10</a></i>
;<br/><a NAME="E6:81_6_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><a NAME="E7:81_6_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:81"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E2:81_6_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E8:81_6_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><a NAME="E9:81_6_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="sppol_2.html#E57:81"><i><font color="Green">E98</font></i></a></i>;<br/><a NAME="E10:81_6_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E11:81_6_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E54:81"><i><font color="Green">E96</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_6_1_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_6_1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:81_6_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 ;<br/><a NAME="E3:81_6_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <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="E4:81_6_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_6"><i><font color="Green">E101</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E5:81_6_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_6_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><b>hence </b><a NAME="E6:81_6_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_6_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:81_6"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</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"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_7_1"/>
( (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_7_1_1"><b>suppose </b></a><a NAME="E1:81_7_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:81_7_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>hence </b><a NAME="E3:81_7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i><a class="ref" href="sppol_2.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_7_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:81_7_1_2"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:81_7_1_2"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E3:81_7_1_2"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_7_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:81_7_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="sppol_2.html#E44:81"><i><font color="Green">E88</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:81_7_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E7:81_7_1_2"/><i><font color="Green">E106</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_7_1_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_7_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:81_7_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_7_1_2"><i><font color="Green">E105</font></i></a></i>;<br/><a NAME="E9:81_7_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:81_7_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E10:81_7_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E53:81"><i><font color="Green">E95</font></i></a>, <a class="txt" href="sppol_2.html#E58:81"><i><font color="Green">E99</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_7_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_7_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="sppol_2.html#E8:81_7_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E11:81_7_1_2"/>
 <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_7_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_5.html#T15">FINSEQ_5:15</a></i>;<br/><b>hence </b><a NAME="E12:81_7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:81_7_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sppol_2.html#E10:81_7_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="sppol_2.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_7_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:81_7_1_3"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:81_7_1_3"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E3:81_7_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_7_1_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_5.html#T15">FINSEQ_5:15</a></i>;<br/><b>hence </b><a NAME="E4:81_7_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i><a class="txt" href="sppol_2.html#E50:81"><i><font color="Green">E92</font></i></a>, <a class="txt" href="sppol_2.html#E59:81"><i><font color="Green">E100</font></i></a>, <a class="txt" href="sppol_2.html#E2:81_7_1_3"><i><font color="Green">E103</font></i></a>, <a class="ref" href="sppol_2.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_7_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:81_7_1_4"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:81_7_1_4"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E3:81_7_1_4"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_7_1_4"><i><font color="Green">E102</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:81_7_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="sppol_2.html#E44:81"><i><font color="Green">E88</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:81_7_1_4"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E7:81_7_1_4"/><i><font color="Green">E106</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_7_1_4"><i><font color="Green">E104</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_7_1_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:81_7_1_4"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_7_1_4"><i><font color="Green">E105</font></i></a></i>;<br/><a NAME="E9:81_7_1_4"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:81_7_1_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E10:81_7_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E53:81"><i><font color="Green">E95</font></i></a>, <a class="txt" href="sppol_2.html#E58:81"><i><font color="Green">E99</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_7_1_4"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_7_1_4"><i><font color="Green">E106</font></i></a>, <a class="txt" href="sppol_2.html#E8:81_7_1_4"><i><font color="Green">E107</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><b>hence </b><a NAME="E11:81_7_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i><a class="txt" href="sppol_2.html#E50:81"><i><font color="Green">E92</font></i></a>, <a class="txt" href="sppol_2.html#E59:81"><i><font color="Green">E100</font></i></a>, <a class="txt" href="sppol_2.html#E2:81_7_1_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_7_1_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="sppol_2.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E61:81"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font> is <a href="topreal1.html#V3">s.n.c.</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="sppol_2.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E62:81"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="topreal1.html#V3">s.n.c.</a>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E63:81"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E65"><i><font color="Green">Lemma59</font></i></a></i>;<br/>
<a NAME="E64:81"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E65:81"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E24"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E66:81"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E5:81"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E64:81"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E67:81"/><b>then </b><i><font color="Green">E104</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/>
<div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:81_8"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E2:81_8"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E3:81_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_8"><i><font color="Green">E106</font></i></a></i>;<br/><a NAME="E4:81_8"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>14</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:81_8"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E6:81_8"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E58:81"><i><font color="Green">E99</font></i></a>, <a class="txt" href="sppol_2.html#E4:81_8"><i><font color="Green">E107</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E106</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:81_9"/><i><font color="Green">E107</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E2:81_9"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:81_9"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><a NAME="E4:81_9"/><b>then </b><i><font color="Green">E109</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E57:81"><i><font color="Green">E98</font></i></a></i>;<br/><a NAME="E5:81_9"/><i><font color="Green">E110</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E58:81"><i><font color="Green">E99</font></i></a>, <a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_9"><i><font color="Green">E108</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E7:81_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E111</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="sppol_2.html#T10" target="_self">Th10</a></i>
;<br/><div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_9_2"><b>now </b></a><div class="add"><a NAME="E1:81_9_2"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E2:81_9_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:81_9_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font>
 ;<br/><a NAME="E5:81_9_2"/><b>then </b><i><font color="Green">E115</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_9_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:81_9_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E7:81_9_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_9_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="sppol_2.html#E5:81_9_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E8:81_9_2"/><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><i><font color="Green">E118</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E8:81_9_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_5.html#T53">FINSEQ_5:53</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_9_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_9_2_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_9_2_2_1"><b>suppose </b></a><a NAME="E1:81_9_2_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E119</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E9:81_9_2"><i><font color="Green">E118</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font>

;<br/><a NAME="E3:81_9_2_2_1"/><i><font color="Green">E120</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E4:81_9_2_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:81_9_2_2_1"/><b>then </b><i><font color="Green">E121</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_9_2_2_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:81_9_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_9_2_2_1"><i><font color="Green">E119</font></i></a></i>;<br/><a NAME="E7:81_9_2_2_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font></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">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_9_2_2_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="sppol_2.html#E5:81_9_2_2_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E8:81_9_2_2_1"/><i><font color="Green">E123</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_9_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:81_9_2_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_9_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:81_9_2_2_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81_9_2_2_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:81_9_2_2_1"/><i><font color="Green">E125</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E12:81_9_2_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E13:81_9_2_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81_9_2_2_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>assume </b><a NAME="E14:81_9_2_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E15:81_9_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81_9"><i><font color="Green">E111</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_9_2_2_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="sppol_2.html#E13:81_9_2_2_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E16:81_9_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E10:81_9_2_2_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="sppol_2.html#E11:81_9_2_2_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E18:81_9_2_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_9_2_2_2"><b>suppose </b></a><a NAME="E1:81_9_2_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:81_9_2_2_2"/><b>then </b>
2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81_9"><i><font color="Green">E107</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:81_9_2_2_2"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:81_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E5:81_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font> <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">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 2
 ;<br/><a NAME="E6:81_9_2_2_2"/><b>then </b><i><font color="Green">E119</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">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E7:81_9_2_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:81_9_2"><i><font color="Green">E118</font></i></a></i>;<br/><a NAME="E8:81_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:81_9_2_2_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:81_9_2_2_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E10:81_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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="E11:81_9_2_2_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81_9"><i><font color="Green">E111</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_9_2"><i><font color="Green">E116</font></i></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><b>end;</b></div><a NAME="E10:81_9"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81_9"><i><font color="Green">E111</font></i></a>, <a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><a NAME="E11:81_9"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:81"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E5:81_9"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_9_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:81_9_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/><a NAME="E2:81_9_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E11:81_9"><i><font color="Green">E113</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_9_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:81_9_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_9"><i><font color="Green">E109</font></i></a>, <a class="txt" href="sppol_2.html#E9:81_9"><i><font color="Green">E112</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_9_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E13:81_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font></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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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#D1">XBOOLE_0:def 1</a></i>;<br/><b>hence </b><a NAME="E14:81_9"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_10"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:81_10"/><i><font color="Green">E107</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:81_10"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:81_10"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:81_10"/><i><font color="Green">E110</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_10"><i><font color="Green">E108</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E7:81_10"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E8:81_10"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E51:81"><i><font color="Green">E93</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E9:81_10"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:81_10"><i><font color="Green">E109</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_10"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:81_10"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><a NAME="E11:81_10"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:81"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_10_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> or <font color="Maroon">c<sub>14</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_10_1_1"><b>suppose </b></a><a NAME="E1:81_10_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:81_10_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:81"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E3:81_10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></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">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_10"><i><font color="Green">E107</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_10_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:81_10_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:81_10"><i><font color="Green">E111</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:81_10_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E9:81"><i><font color="Green">E73</font></i></a>, <a class="txt" href="sppol_2.html#E56:81"><i><font color="Green">E97</font></i></a>, <a class="txt" href="sppol_2.html#E11:81_10"><i><font color="Green">E113</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_10_1_2"><b>suppose </b></a><a NAME="E1:81_10_1_2"/>
<font color="Maroon">c<sub>14</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:81_10_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_10"><i><font color="Green">E108</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:81_10_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:81_10"><i><font color="Green">E112</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E71:81"/><b>then </b>
( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E57:81"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E72:81"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E67:81"><i><font color="Green">E104</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E73:81"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="topreal1.html#V3">s.n.c.</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E62:81"><i><font color="Green">E102</font></i></a>, <a class="txt" href="sppol_2.html#E68:81"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sppol_2.html#E69:81"><i><font color="Green">E106</font></i></a>, <a class="ref" href="sppol_2.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E74:81"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="sppol_2.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E75:81"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E76:81"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E43:81"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E51:81"><i><font color="Green">E93</font></i></a>, <a class="txt" href="sppol_2.html#E53:81"><i><font color="Green">E95</font></i></a>, <a class="txt" href="sppol_2.html#E54:81"><i><font color="Green">E96</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/>
<a NAME="E77:81"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E75:81"><i><font color="Green">E108</font></i></a>, <a class="txt" href="sppol_2.html#E56"><i><font color="Green">Lemma53</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sppol_2.html#E36:81"><i><font color="Green">E83</font></i></a>, <a class="txt" href="sppol_2.html#E46:81"><i><font color="Green">E89</font></i></a>, <a class="txt" href="sppol_2.html#E60:81"><i><font color="Green">E101</font></i></a>, <a class="txt" href="sppol_2.html#E73:81"><i><font color="Green">E107</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>thus </b><a NAME="E79:81"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T47">FINSEQ_5:47</a></i>;<br/>

<a NAME="E80:81"/><i><font color="Green">E110</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><i><font color="Green">E111</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E47:81"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E47:81"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E2:81"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sppol_2.html#E47:81"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
;<br/><br/>

<b>thus </b><i><font color="Green">E112</font></i>: <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T48">FINSEQ_5:48</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T45">FINSEQ_5:45</a></i>
;<br/>

<a NAME="E83:81"/><i><font color="Green">E113</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E84:81"/><i><font color="Green">E114</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><i><font color="Green">E115</font></i>: <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E41:81"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E83:81"><i><font color="Green">E113</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="finseq_5.html#T56">FINSEQ_5:56</a></i>
;<br/><br/>

<a NAME="E86:81"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E87:81"/><b>then </b><i><font color="Green">E116</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E88:81"/><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E89:81"/><i><font color="Green">E118</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<i><font color="Green">E119</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E4:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E51:81"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_5.html#T24">FINSEQ_5:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E4:81"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E47:81"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sppol_2.html#E54:81"><i><font color="Green">E96</font></i></a>, <a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E41:81"><i><font color="Green">E86</font></i></a>, <a class="txt" href="sppol_2.html#E48:81"><i><font color="Green">E91</font></i></a>, <a class="ref" href="sppol_2.html#T23" target="_self">Th23</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>
;<br/>
<b>thus </b><a NAME="E91:81"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_15"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:81_15_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_15_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="sppol_2.html#E1:81_15_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_1_1_1"><b>suppose </b></a><a NAME="E1:81_15_1_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_15_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E79:81"><i><font color="Green">E109</font></i></a>, <a class="txt" href="sppol_2.html#E80:81"><i><font color="Green">E110</font></i></a>, <a class="txt" href="sppol_2.html#E81:81"><i><font color="Green">E111</font></i></a>, <a class="txt" href="sppol_2.html#E89:81"><i><font color="Green">E118</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E3:81_15_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E87:81"><i><font color="Green">E116</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_1_1_1"><i><font color="Green">E121</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="81_15_1_1_2"><b>suppose </b></a><a NAME="E1:81_15_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_15_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E82:81"><i><font color="Green">E112</font></i></a>, <a class="txt" href="sppol_2.html#E84:81"><i><font color="Green">E114</font></i></a>, <a class="txt" href="sppol_2.html#E85:81"><i><font color="Green">E115</font></i></a>, <a class="txt" href="sppol_2.html#E88:81"><i><font color="Green">E117</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E3:81_15_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E87:81"><i><font color="Green">E116</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_1_1_2"><i><font color="Green">E121</font></i></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>
<b>end;</b></div>

<a NAME="E2:81_15"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E90:81"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E3:81_15"/><i><font color="Green">E121</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T45">FINSEQ_5:45</a></i>;<br/>
<a NAME="E4:81_15"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E5:81_15"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<i><font color="Green">E124</font></i>: <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E4:81_15"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_15_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_3_1"><b>suppose </b></a><a NAME="E1:81_15_3_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_15_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_15_3_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:81_15_3_1_1"/><i><font color="Green">E126</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E4:81_15_3_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>15</sub></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">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:81_15_3_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:81_15_3_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:81_15_3_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="sppol_2.html#E5:81_15_3_1_1"><i><font color="Green">E128</font></i></a>, <a class="ref" href="sppol_2.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E7:81_15_3_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E8:81_15_3_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E37:81"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E10:81_15_3_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font>
 ;<br/><a NAME="E11:81_15_3_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:81"><i><font color="Green">E74</font></i></a>, <a class="txt" href="sppol_2.html#E37:81"><i><font color="Green">E84</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E12:81_15_3_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E7:81_15_3_1_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:81_15_3_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:81_15_3_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:81_15_3_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E15:81_15_3_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E13:81_15_3_1_1"><i><font color="Green">E132</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E16:81_15_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_15"><i><font color="Green">E124</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_3_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="sppol_2.html#E7:81_15_3_1_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E17:81_15_3_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:81_15_3_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:81_15_3_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_15"><i><font color="Green">E121</font></i></a>, <a class="txt" href="sppol_2.html#E4:81_15_3_1_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_15_3_1_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_15_3_1_1_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_15_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E3:81_15_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></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="sppol_2.html#E3:81_15_3_1_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:81_15_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></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="sppol_2.html#E6:81_15_3_1_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="sppol_2.html#E15:81_15_3_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_3_1_1_1_1"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:81_15_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:81_15_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_15"><i><font color="Green">E121</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_15"><i><font color="Green">E124</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_3_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="sppol_2.html#E4:81_15_3_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="sppol_2.html#E17:81_15_3_1_1"><i><font color="Green">E134</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_3_1_1_1_1"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_15_3_1_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_15_3_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E18:81_15_3_1_1"><i><font color="Green">E135</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:81_15_3_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E4:81_15_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></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="E5:81_15_3_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:81_15_3_1_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="sppol_2.html#E15:81_15_3_1_1"><i><font color="Green">E133</font></i></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><b>end;</b></div><a NAME="E3:81_15_3_1"/><b>then </b><i><font color="Green">E126</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E10:81"><i><font color="Green">E74</font></i></a>, <a class="txt" href="sppol_2.html#E6:81_15"><i><font color="Green">E124</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_3_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="sppol_2.html#T45" target="_self">Th45</a></i>
;<br/><a NAME="E5:81_15_3_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E64"><i><font color="Green">Lemma58</font></i></a></i>;<br/><a NAME="E6:81_15_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:81"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E7:81_15_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/><a NAME="E8:81_15_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81_15_3_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E9:81_15_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:81_15"><i><font color="Green">E120</font></i></a>, <a class="txt" href="sppol_2.html#E4:81_15_3_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E10:81_15_3_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_15_3_2"><b>suppose </b></a><a NAME="E1:81_15_3_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_15_3_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:81"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E5:81_15"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_5.html#T44">FINSEQ_5:44</a></i>;<br/><b>then </b><i><font color="Green">E126</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_3_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="finseq_5.html#T53">FINSEQ_5:53</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 1

;<br/><i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="sppol_2.html#E3:81_15_3_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="topreal1.html#T28">TOPREAL1:28</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/><a NAME="E5:81_15_3_2"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E64"><i><font color="Green">Lemma58</font></i></a></i>;<br/><b>hence </b><a NAME="E6:81_15_3_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:81"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E2:81_15"><i><font color="Green">E120</font></i></a>, <a class="txt" href="sppol_2.html#E1:81_15_3_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="sppol_2.html#E4:81_15_3_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E6:81"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E8:81"><i><font color="Green">E72</font></i></a>, <a class="ref" href="sppol_2.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E90:81"><i><font color="Green">E119</font></i></a>, <a class="ref" href="sppol_2.html#T22" target="_self">Th22</a></i>
;<br/>


</div>
