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

<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font>, <font color="Maroon" title="c3">g1</font> be  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font> is  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> &amp; <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a>  )
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green" title="E43">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:43"/><i><font color="Green" title="E44">A3</font></i>: 
( <font color="Maroon" title="c3">g1</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c3">g1</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">g1</font> = <font color="Maroon" title="c3">g1</font> as  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan12.html#E3:43"><i><font color="Green" title="E44">A3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">Lf</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span>;<br/>
<a NAME="E5:43"/>
<font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font> <a href="jordan12.html#R1" title="JORDAN12:pred.1">is_in_general_position_wrt</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="jordan12.html#D2" target="_self" title="JORDAN12:def.2">Def2</a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green" title="E45">A4</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="ref" href="jordan12.html#D1" target="_self" title="JORDAN12:def.1">Def1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font> implies for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> holds <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b2">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> ) ) );<br/>
<a NAME="E7:43"/><i><font color="Green" title="E46">A5</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/>
<a NAME="E8:43"/><i><font color="Green" title="E47">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:43"/><i><font color="Green" title="E48">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:43"/><b>then </b><i><font color="Green" title="E49">A8</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:43"/><i><font color="Green" title="E50">A9</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c6">a</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E51">A10</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span>
 ;<br/><a NAME="E2:43_1"/>
<font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E3:43_1"/><b>then </b><i><font color="Green" title="E52">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan12.html#E9:43"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><i><font color="Green" title="E53">A12</font></i>:  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">a</font> = 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E1:43_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E10:43"><i><font color="Green" title="E49">A8</font></i></a>, <a class="txt" href="jordan12.html#E11:43"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="topreal3.html#T45" title="TOPREAL3:th.45">TOPREAL3:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E3:43_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="topreal1.html#T28" title="TOPREAL1:th.28">TOPREAL1:28</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,1

;<br/><a NAME="E5:43_1"/><i><font color="Green" title="E54">A13</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:43_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E7:43_1"/><b>then </b><i><font color="Green" title="E55">A14</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><font color="Maroon" title="c6">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">a</font></span>)</span> = 
<font color="Maroon" title="c6">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="jordan12.html#E1:43_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E1:43_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="jordan12.html#E5:43_1"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E9:43_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c6">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E2:43"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="jordan12.html#E4:43_1"><i><font color="Green" title="E53">A12</font></i></a>, <a class="txt" href="jordan12.html#E7:43_1"><i><font color="Green" title="E55">A14</font></i></a>, <a class="ref" href="jordan12.html#T34" target="_self" title="JORDAN12:th.34">Th34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:43"/><b>then </b><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
;<br/>
<div><i><font color="Green" title="E52">A16</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" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:43_2"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>and </b><br/><a NAME="E2:43_2"/><i><font color="Green" title="E54">A18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/><a NAME="E3:43_2"/><i><font color="Green" title="E55">A19</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:43_2"/><b>then </b><i><font color="Green" title="E56">A20</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E57">A21</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font>
 ;<br/><b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E2:43_2_1"/><i><font color="Green" title="E58">A22</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E3:43_2_1"/><i><font color="Green" title="E59">A23</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:43_2_1"/><i><font color="Green" title="E60">A24</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E4:43_2"><i><font color="Green" title="E56">A20</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:43_2_1"/><i><font color="Green" title="E61">A25</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g1</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E8:43"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2"><i><font color="Green" title="E56">A20</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:43_2_1"/><i><font color="Green" title="E62">A26</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E4:43_2"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:43_2_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E4:43_2"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:43_2_1"/><b>then </b><i><font color="Green" title="E63">A27</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E9:43_2_1"/><i><font color="Green" title="E64">A28</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E7:43"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="jordan12.html#T9" target="_self" title="JORDAN12:th.9">Th9</a></i>;<br/><a NAME="E10:43_2_1"/><i><font color="Green" title="E65">A29</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="jordan12.html#T9" target="_self" title="JORDAN12:th.9">Th9</a></i>;<br/><a NAME="E11:43_2_1"/><i><font color="Green" title="E66">A30</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="jordan12.html#T9" target="_self" title="JORDAN12:th.9">Th9</a></i>;<br/><a NAME="E12:43_2_1"/><i><font color="Green" title="E67">A31</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><i><font color="Green" title="E68">A32</font></i>: <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="jordan12.html#E6:43_2_1"><i><font color="Green" title="E62">A26</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>
;<br/><a NAME="E14:43_2_1"/><i><font color="Green" title="E69">A33</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">a</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="jordan12.html#T7" target="_self" title="JORDAN12:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/><a NAME="E16:43_2_1"/><i><font color="Green" title="E70">A34</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c8">b</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="jordan12.html#T7" target="_self" title="JORDAN12:th.7">Th7</a></i>;<br/><a NAME="E17:43_2_1"/><i><font color="Green" title="E71">A35</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E18:43_2_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E19:43_2_1"/><b>then </b><i><font color="Green" title="E72">A36</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><i><font color="Green" title="E73">A37</font></i>: <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font></span>)</span> = 
<font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E3:43_2_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E17:43_2_1"><i><font color="Green" title="E71">A35</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>set </b><font color="Maroon" title="c9">c</font> =  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">s1</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="jordan12.html#E16:43_2_1"><i><font color="Green" title="E70">A34</font></i></a>, <a class="ref" href="jordan12.html#T12" target="_self" title="JORDAN12:th.12">Th12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">s2</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="jordan12.html#T13" target="_self" title="JORDAN12:th.13">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">s</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="jordan12.html#E14:43_2_1"><i><font color="Green" title="E69">A33</font></i></a>, <a class="ref" href="jordan12.html#T12" target="_self" title="JORDAN12:th.12">Th12</a></i>;<br/><a NAME="E24:43_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><b>then </b> <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E5:43_2_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="topreal3.html#T45" title="TOPREAL3:th.45">TOPREAL3:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E26:43_2_1"/><b>then </b><i><font color="Green" title="E74">A38</font></i>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">s2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/><a NAME="E27:43_2_1"/><i><font color="Green" title="E75">A39</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E28:43_2_1"/><i><font color="Green" title="E76">A40</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E29:43_2_1"/><i><font color="Green" title="E77">A41</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E30:43_2_1"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E31:43_2_1"/><b>then </b><i><font color="Green" title="E78">A42</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E5:43_2_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="jordan12.html#E28:43_2_1"><i><font color="Green" title="E76">A40</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E32:43_2_1"/><i><font color="Green" title="E79">A43</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E29:43_2_1"><i><font color="Green" title="E77">A41</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><i><font color="Green" title="E80">A44</font></i>: <font color="Maroon" title="c7">a</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E32:43_2_1"><i><font color="Green" title="E79">A43</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
;<br/><a NAME="E34:43_2_1"/><i><font color="Green" title="E81">A45</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E8:43"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="jordan12.html#E6:43_2_1"><i><font color="Green" title="E62">A26</font></i></a>, <a class="txt" href="jordan12.html#E28:43_2_1"><i><font color="Green" title="E76">A40</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E35:43_2_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43_2"><i><font color="Green" title="E53">A17</font></i></a></i>;<br/><a NAME="E36:43_2_1"/><b>then </b><i><font color="Green" title="E82">A46</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E83">A47</font></i>: <font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E31:43_2_1"><i><font color="Green" title="E78">A42</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="jordan12.html#E31:43_2_1"><i><font color="Green" title="E78">A42</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="jordan12.html#E5:43_2_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><i><font color="Green" title="E84">A48</font></i>: <font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E34:43_2_1"><i><font color="Green" title="E81">A45</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E2:43_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="jordan12.html#E34:43_2_1"><i><font color="Green" title="E81">A45</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan12.html#E8:43"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E39:43_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E12:43_2_1"><i><font color="Green" title="E67">A31</font></i></a>, <a class="txt" href="jordan12.html#E27:43_2_1"><i><font color="Green" title="E75">A39</font></i></a>, <a class="txt" href="jordan12.html#E36:43_2_1"><i><font color="Green" title="E82">A46</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">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" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E12:43_2_1"><i><font color="Green" title="E67">A31</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">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" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E33:43_2_1"><i><font color="Green" title="E80">A44</font></i></a>, <a class="txt" href="jordan12.html#E37:43_2_1"><i><font color="Green" title="E83">A47</font></i></a>, <a class="txt" href="jordan12.html#E38:43_2_1"><i><font color="Green" title="E84">A48</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">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" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E43:43_2_1"/><b>then </b><i><font color="Green" title="E85">A49</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E5:43_2_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E44:43_2_1"/><i><font color="Green" title="E86">A50</font></i>: 
<font color="Maroon" title="c10">s1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">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" title="c10">s1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">s2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_2_1_7"/><i><font color="Green" title="E87">A51</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">s1</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">s2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:43_2_1_7"/><i><font color="Green" title="E88">A52</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E3:43_2_1_7"><i><font color="Green" title="E87">A51</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:43_2_1_7"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:43_2_1_7"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E43:43_2_1"><i><font color="Green" title="E85">A49</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:43_2_1_7"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E5:43_2_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">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#E6:43"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1_7"><i><font color="Green" title="E88">A52</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">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" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">s1</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">s1</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E87">A53</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c10">s1</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c13">c1</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span></span>)</span> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="int_1.html#NM1" title="INT_1:NM.1">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" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">s2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">s2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E88">A54</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">s2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">c2</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">s</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_1_1_1_1"/><i><font color="Green" title="E89">A55</font></i>: 
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">c2</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E26:43_2_1"><i><font color="Green" title="E74">A38</font></i></a>, <a class="txt" href="jordan12.html#E44:43_2_1"><i><font color="Green" title="E86">A50</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_1_1_1_1"/><i><font color="Green" title="E90">A56</font></i>: 
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_1_1_1_1"><i><font color="Green" title="E88">A54</font></i></a>, <a class="ref" href="jordan12.html#T34" target="_self" title="JORDAN12:th.34">Th34</a></i>;<br/><a NAME="E6:43_2_1_8_1_1_1_1"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_1"><i><font color="Green" title="E87">A53</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_1_1_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E8:43_2_1"><i><font color="Green" title="E63">A27</font></i></a>, <a class="txt" href="jordan12.html#E13:43_2_1"><i><font color="Green" title="E68">A32</font></i></a>, <a class="txt" href="jordan12.html#E19:43_2_1"><i><font color="Green" title="E72">A36</font></i></a>, <a class="txt" href="jordan12.html#E20:43_2_1"><i><font color="Green" title="E73">A37</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1_8_1_1_1_1"><i><font color="Green" title="E89">A55</font></i></a>, <a class="txt" href="jordan12.html#E5:43_2_1_8_1_1_1_1"><i><font color="Green" title="E90">A56</font></i></a>, <a class="ref" href="jordan12.html#T17" target="_self" title="JORDAN12:th.17">Th17</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" title="E88">A57</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">s2</font> is  not  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">c2</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">s</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_1_1_1_2"/><i><font color="Green" title="E89">A58</font></i>: 
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">c2</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E26:43_2_1"><i><font color="Green" title="E74">A38</font></i></a>, <a class="txt" href="jordan12.html#E44:43_2_1"><i><font color="Green" title="E86">A50</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_1_1_1_2"/><i><font color="Green" title="E90">A59</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_1_1_1_2"><i><font color="Green" title="E88">A57</font></i></a>, <a class="ref" href="jordan12.html#T34" target="_self" title="JORDAN12:th.34">Th34</a></i>;<br/><a NAME="E6:43_2_1_8_1_1_1_2"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_1"><i><font color="Green" title="E87">A53</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_1_1_1_2"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E8:43_2_1"><i><font color="Green" title="E63">A27</font></i></a>, <a class="txt" href="jordan12.html#E13:43_2_1"><i><font color="Green" title="E68">A32</font></i></a>, <a class="txt" href="jordan12.html#E19:43_2_1"><i><font color="Green" title="E72">A36</font></i></a>, <a class="txt" href="jordan12.html#E20:43_2_1"><i><font color="Green" title="E73">A37</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1_8_1_1_1_2"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E5:43_2_1_8_1_1_1_2"><i><font color="Green" title="E90">A59</font></i></a>, <a class="ref" href="jordan12.html#T17" target="_self" title="JORDAN12:th.17">Th17</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" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="E87">A60</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">s1</font> is  not  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c13">c1</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">b</font></span>)</span></span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="int_1.html#NM1" title="INT_1:NM.1">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" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">s2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">s2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E88">A61</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">s2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">c2</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">s</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_2_1_1_1"/><i><font color="Green" title="E89">A62</font></i>: 
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">c2</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E26:43_2_1"><i><font color="Green" title="E74">A38</font></i></a>, <a class="txt" href="jordan12.html#E44:43_2_1"><i><font color="Green" title="E86">A50</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_2_1_1_1"/><i><font color="Green" title="E90">A63</font></i>: 
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_2_1_1_1"><i><font color="Green" title="E88">A61</font></i></a>, <a class="ref" href="jordan12.html#T34" target="_self" title="JORDAN12:th.34">Th34</a></i>;<br/><a NAME="E6:43_2_1_8_2_1_1_1"/>
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_2"><i><font color="Green" title="E87">A60</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_2_1_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E8:43_2_1"><i><font color="Green" title="E63">A27</font></i></a>, <a class="txt" href="jordan12.html#E13:43_2_1"><i><font color="Green" title="E68">A32</font></i></a>, <a class="txt" href="jordan12.html#E19:43_2_1"><i><font color="Green" title="E72">A36</font></i></a>, <a class="txt" href="jordan12.html#E20:43_2_1"><i><font color="Green" title="E73">A37</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1_8_2_1_1_1"><i><font color="Green" title="E89">A62</font></i></a>, <a class="txt" href="jordan12.html#E5:43_2_1_8_2_1_1_1"><i><font color="Green" title="E90">A63</font></i></a>, <a class="ref" href="jordan12.html#T17" target="_self" title="JORDAN12:th.17">Th17</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" title="E88">A64</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">s2</font> is  not  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">c2</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">s</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E4:43_2_1_8_2_1_1_2"/><i><font color="Green" title="E89">A65</font></i>: 
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">c2</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E26:43_2_1"><i><font color="Green" title="E74">A38</font></i></a>, <a class="txt" href="jordan12.html#E44:43_2_1"><i><font color="Green" title="E86">A50</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/><a NAME="E5:43_2_1_8_2_1_1_2"/><i><font color="Green" title="E90">A66</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:43_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_2_1_1_2"><i><font color="Green" title="E88">A64</font></i></a>, <a class="ref" href="jordan12.html#T34" target="_self" title="JORDAN12:th.34">Th34</a></i>;<br/><a NAME="E6:43_2_1_8_2_1_1_2"/>
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="jordan12.html#E1:43_2_1_8_2"><i><font color="Green" title="E87">A60</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E7:43_2_1_8_2_1_1_2"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:43"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan12.html#E8:43_2_1"><i><font color="Green" title="E63">A27</font></i></a>, <a class="txt" href="jordan12.html#E9:43_2_1"><i><font color="Green" title="E64">A28</font></i></a>, <a class="txt" href="jordan12.html#E10:43_2_1"><i><font color="Green" title="E65">A29</font></i></a>, <a class="txt" href="jordan12.html#E11:43_2_1"><i><font color="Green" title="E66">A30</font></i></a>, <a class="txt" href="jordan12.html#E13:43_2_1"><i><font color="Green" title="E68">A32</font></i></a>, <a class="txt" href="jordan12.html#E19:43_2_1"><i><font color="Green" title="E72">A36</font></i></a>, <a class="txt" href="jordan12.html#E20:43_2_1"><i><font color="Green" title="E73">A37</font></i></a>, <a class="txt" href="jordan12.html#E4:43_2_1_8_2_1_1_2"><i><font color="Green" title="E89">A65</font></i></a>, <a class="txt" href="jordan12.html#E5:43_2_1_8_2_1_1_2"><i><font color="Green" title="E90">A66</font></i></a>, <a class="ref" href="jordan12.html#T18" target="_self" title="JORDAN12:th.18">Th18</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" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E15:43"/><i><font color="Green" title="E53">A67</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="jordan12.html#E13:43"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="jordan12.html#E14:43"><i><font color="Green" title="E52">A16</font></i></a>);<br/></i>
<a NAME="E16:43"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:43"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E18:43"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</font></span>)</span></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:43"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="jordan12.html#E15:43"><i><font color="Green" title="E53">A67</font></i></a></i>;<br/>


</div>
