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

<b>let </b><font color="Maroon">f1</font> be  <a href="topreal1.html#V1">special</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">f2</font> be  <a href="topreal1.html#V1">special</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">g1</font> be  <a href="topreal1.html#V1">special</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font> is  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a> &amp; <font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font>,<font color="Maroon">g1</font> <a href="jordan12.html#R2" target="_self">are_in_general_position</a>  )
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:43"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">g1</font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">g1</font> is <a href="topreal1.html#V3">s.n.c.</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font> as  <a href="topreal1.html#V1">special</a> <a href="topreal1.html#V2">unfolded</a> <a href="topreal1.html#V3">s.n.c.</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Lf</font> =  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span>;<br/>
<a NAME="E5:43"/>
<font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> implies for <font color="Olive">a</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/>(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) ) );<br/>
<a NAME="E7:43"/><i><font color="Green">E55</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">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E8:43"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</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">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:43"/><i><font color="Green">E57</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:43"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:43"/><i><font color="Green">E59</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 2</span>)</span>
 ;<br/><a NAME="E2:43_1"/>
<font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E3:43_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><i><font color="Green">E62</font></i>:  <a href="topreal1.html#K5">L~</a> <font color="Maroon">a</font> = 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g1</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>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, <a class="ref" href="topreal3.html#T45">TOPREAL3:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="ref" href="topreal1.html#T28">TOPREAL1:28</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,1

;<br/><a NAME="E5:43_1"/><i><font color="Green">E63</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:43_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E7:43_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><font color="Maroon">a</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">a</font></span>)</span> = 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan12.html#E6:43"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E9:43_1"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E2:43"><i><font color="Green">E43</font></i></a>, , <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:43"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
;<br/>
<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:43_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>and </b><br/><a NAME="E2:43_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><a NAME="E3:43_2"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:43_2"/><b>then </b><i><font color="Green">E78</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">k</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">g1</font>
 ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E2:43_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E3:43_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:43_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E8:43"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:43_2_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E8:43"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E11:43"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:43_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E8:43"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:43_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E8:43"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:43_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E9:43_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 1 <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, , </i>;<br/><a NAME="E10:43_2_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E11:43_2_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E12:43_2_1"/><i><font color="Green">E144</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><i><font color="Green">E145</font></i>: <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">a</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">a</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>
;<br/><a NAME="E14:43_2_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font>,<font color="Maroon">a</font> <a href="jordan12.html#R2" target="_self">are_in_general_position</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E16:43_2_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font>,<font color="Maroon">b</font> <a href="jordan12.html#R2" target="_self">are_in_general_position</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E17:43_2_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E18:43_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E19:43_2_1"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><i><font color="Green">E150</font></i>: <font color="Maroon">b</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">b</font></span>)</span> = 
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E11:43"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>set </b><font color="Maroon">c</font> =  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">k</font>;<br/><b>reconsider </b><font color="Maroon">s1</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</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> <font color="Maroon">b</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, </i>;<br/><b>reconsider </b><font color="Maroon">s2</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></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">g1</font>,<font color="Maroon">k</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</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> <font color="Maroon">a</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green">E69</font></i></a>, </i>;<br/><a NAME="E24:43_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><b>then </b> <a href="topreal1.html#K5">L~</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="topreal3.html#T45">TOPREAL3:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E26:43_2_1"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><a NAME="E27:43_2_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E28:43_2_1"/><i><font color="Green">E153</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E29:43_2_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E30:43_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E31:43_2_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="jordan12.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E32:43_2_1"/><i><font color="Green">E162</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><i><font color="Green">E163</font></i>: <font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
;<br/><a NAME="E34:43_2_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, , , <a class="ref" href="jordan12.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E35:43_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E36:43_2_1"/><b>then </b><i><font color="Green">E165</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E166</font></i>: <font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E4:43_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><i><font color="Green">E167</font></i>: <font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E39:43_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></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">a</font>,<font color="Maroon">k</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">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, <a class="ref" href="goboard2.html#T9">GOBOARD2:9</a></i>;<br/><a NAME="E40:43_2_1"/><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">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</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">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E41:43_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">b</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"><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</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">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, , , <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E42:43_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">b</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">g1</font>,<font color="Maroon">k</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">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E43:43_2_1"/><b>then </b><i><font color="Green">E168</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">b</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">g1</font>,<font color="Maroon">k</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">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E44:43_2_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">s1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2_1_7"/>
<font color="Maroon">s1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">s2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_2_1_7"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:43_2_1_7"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:43_2_1_7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">b</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">g1</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:43_2_1_7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:43_2_1_7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E8:43_2_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan12.html#E3:43"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1_8"/>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">s1</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  or  not  <a href="card_1.html#K4">card</a> <font color="Maroon">s1</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_1"><b>suppose </b></a><a NAME="E1:43_2_1_8_1"/><i><font color="Green">E172</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">s1</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c1</font> =  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">b</font></span>)</span></span>)</span> as  <a href="abian.html#V1">even</a> <a href="int_1.html#NM1">Integer</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_8_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1_8_1_1_1"/>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">s2</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  or  not  <a href="card_1.html#K4">card</a> <font color="Maroon">s2</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_2_1_8_1_1_1_1"/><i><font color="Green">E173</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">s2</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c2</font> =  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> as  <a href="abian.html#V1">even</a> <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">s</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_1_1_1_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_1_1_1_1"/><i><font color="Green">E175</font></i>: 
 ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E6:43_2_1_8_1_1_1_1"/>
 ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">b</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">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E6:43"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_1_1_1_1"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green">E68</font></i></a>, , , <a class="txt" href="jordan12.html#E3:43_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1"><i><font color="Green">E90</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_2_1_8_1_1_1_2"/><i><font color="Green">E176</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">s2</font> is  not  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c2</font> =  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> as  <a href="abian.html#V1">odd</a> <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">s</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_1_1_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_1_1_1_2"/><i><font color="Green">E178</font></i>: 
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E6:43_2_1_8_1_1_1_2"/>
 ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">b</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">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E6:43"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_1_1_1_2"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green">E68</font></i></a>, , , <a class="txt" href="jordan12.html#E3:43_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1"><i><font color="Green">E90</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:43_2_1_8_1"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_2"><b>suppose </b></a><a NAME="E1:43_2_1_8_2"/><i><font color="Green">E179</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">s1</font> is  not  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c1</font> =  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">b</font></span>)</span></span>)</span> as  <a href="abian.html#V1">odd</a> <a href="int_1.html#NM1">Integer</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_8_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1_8_2_1_1"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">s2</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  or  not  <a href="card_1.html#K4">card</a> <font color="Maroon">s2</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_2_1_8_2_1_1_1"/><i><font color="Green">E180</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">s2</font> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c2</font> =  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> as  <a href="abian.html#V1">even</a> <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">s</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_2_1_1_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_2_1_1_1"/><i><font color="Green">E182</font></i>: 
 ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E6:43_2_1_8_2_1_1_1"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">b</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">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E6:43"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_2_1_1_1"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green">E68</font></i></a>, , , <a class="txt" href="jordan12.html#E3:43_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1"><i><font color="Green">E90</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_8_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_2_1_8_2_1_1_2"/><i><font color="Green">E183</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">s2</font> is  not  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c2</font> =  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> as  <a href="abian.html#V1">odd</a> <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">s</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_2_1_1_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_2_1_1_2"/><i><font color="Green">E185</font></i>: 
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E6:43_2_1_8_2_1_1_2"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">b</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">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E6:43"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E9:43"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_2_1_1_2"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green">E42</font></i></a>, , , <a class="txt" href="jordan12.html#E13:43"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan12.html#E14:43"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green">E68</font></i></a>, , , <a class="txt" href="jordan12.html#E3:43_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1"><i><font color="Green">E90</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:43_2_1_8_2"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:43_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E15:43"/><i><font color="Green">E186</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(<a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan12.html#T5" target="_self">Th5</a>);<br/></i>
<a NAME="E16:43"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</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">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:43"/><b>then </b>
<font color="Maroon">g1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E18:43"/>
(  <a href="card_1.html#K1">Card</a> <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">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></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">g1</font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  iff  ex <font color="Olive">C</font> being  <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> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">g1</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">g1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green">E43</font></i></a>, </i>;<br/>


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