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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp;  <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 &amp; <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>  implies (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:42"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 <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
 <span class="kw">and </span><br/><a NAME="E4:42"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> ) )</span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E44">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Lf</font> =  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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="E6:42"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="jordan12.html#D2" target="_self" title="JORDAN12:def.2">Def2</a></span>;<br/>
<a NAME="E7:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan12.html#D1" target="_self" title="JORDAN12:def.1">Def1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $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="partfun1.html#K2" title="PARTFUN1:func.2">|</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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="E8:42"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:42"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E43">A3</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E48">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <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] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E2:42_1"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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]</span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E4:42_1"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A13</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><a NAME="E2:42_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E3:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A14</font></span>: 
<font color="Maroon" title="c7">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">s1</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="jordan12.html#T6" target="_self" title="JORDAN12:th.6">Th7</a>, <a class="ref" href="jordan12.html#T11" target="_self" title="JORDAN12:th.11">Th12</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">c</font> =  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font>);<br/><a NAME="E5:42_1_1"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">s2</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K2" title="TOPREAL1:func.2">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="jordan12.html#T12" target="_self" title="JORDAN12:th.12">Th13</a></span>;<br/><a NAME="E7:42_1_1"/><span class="lab"><font color="Green" title="E56">A16</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A17</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><a NAME="E9:42_1_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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>;<br/><a NAME="E10:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A18</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:42_1_1"/><span class="lab"><font color="Green" title="E59">A19</font></span>: 
<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#K3" title="TOPREAL1:func.3">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="jordan12.html#T8" target="_self" title="JORDAN12:th.8">Th9</a></span>;<br/><span class="kw">assume </span><a NAME="E12:42_1_1"/><span class="lab"><font color="Green" title="E60">A20</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="partfun1.html#K2" title="PARTFUN1:func.2">|</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> 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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="b2">C</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">b<sub>3</sub></font> &amp; <font color="Olive" title="b2">C</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="b2">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>3</sub></font> ) )</span><br/><a NAME="E13:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A21</font></span>: 
( <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#K3" title="TOPREAL1:func.3">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> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="jordan12.html#T8" target="_self" title="JORDAN12:th.8">Th9</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c11">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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> implies (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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">C</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">b<sub>2</sub></font> &amp; <font color="Olive" title="b1">C</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">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E14:42_1_1"/><span class="lab"><font color="Green" title="E62">A22</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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">C</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">b<sub>2</sub></font> &amp; <font color="Olive" title="b1">C</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">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/><a NAME="E15:42_1_1"/><span class="lab"><font color="Green" title="E63">A23</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><a NAME="E16:42_1_1"/><span class="lab"><font color="Green" title="E64">A24</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A25</font></span>: <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="c11">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c11">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="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
;<br/><a NAME="E18:42_1_1"/><span class="lab"><font color="Green" title="E66">A26</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E19:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A27</font></span>: 
<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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E16:42_1_1"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="txt" href="#E15:42_1_1"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A28</font></span>: <font color="Maroon" title="c11">a</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c11">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="txt" href="#E19:42_1_1"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E18:42_1_1"><span class="lab"><font color="Green" title="E66">A26</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><a NAME="E21:42_1_1"/><span class="lab"><font color="Green" title="E69">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E22:42_1_1"/>
<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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E23:42_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c11">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="c11">a</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E21:42_1_1"><span class="lab"><font color="Green" title="E69">A29</font></span></a>, <a class="txt" href="#E10:42_1_1"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></span>;<br/><a NAME="E24:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A30</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c11">a</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E21:42_1_1"><span class="lab"><font color="Green" title="E69">A29</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E25:42_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E26:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A31</font></span>: 
<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="c11">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">s</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="jordan12.html#T6" target="_self" title="JORDAN12:th.6">Th7</a>, <a class="ref" href="jordan12.html#T11" target="_self" title="JORDAN12:th.11">Th12</a></span>;<br/><a NAME="E28:42_1_1"/><span class="lab"><font color="Green" title="E72">A32</font></span>: 
<font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E29:42_1_1"/><span class="lab"><font color="Green" title="E73">A33</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E30:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A34</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E75">A35</font></span>: <font color="Maroon" title="c11">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="partfun1.html#K2" title="PARTFUN1:func.2">|</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="partfun1.html#K2" title="PARTFUN1:func.2">|</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1_1"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
;<br/><span class="lab"><font color="Green" title="E76">A36</font></span>: <font color="Maroon" title="c7">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="c7">b</font></span>)</span> = 
<font color="Maroon" title="c7">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42_1_1"><span class="lab"><font color="Green" title="E73">A33</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><a NAME="E33:42_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E7:42_1_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E34:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A37</font></span>: 
<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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42_1_1"><span class="lab"><font color="Green" title="E74">A34</font></span></a>, <a class="txt" href="#E15:42_1_1"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">a</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c11">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="txt" href="#E34:42_1_1"><span class="lab"><font color="Green" title="E77">A37</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42_1_1"><span class="lab"><font color="Green" title="E74">A34</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><a NAME="E36:42_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:42_1_1"><span class="lab"><font color="Green" title="E75">A35</font></span></a>, <a class="txt" href="#E20:42_1_1"><span class="lab"><font color="Green" title="E68">A28</font></span></a>, <a class="txt" href="#E24:42_1_1"><span class="lab"><font color="Green" title="E70">A30</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E37:42_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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#K2" title="TOPREAL1:func.2">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E38:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A38</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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#K2" title="TOPREAL1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42_1_1"><span class="lab"><font color="Green" title="E74">A34</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E39:42_1_1"/><span class="lab"><font color="Green" title="E79">A39</font></span>: 
<font color="Maroon" title="c8">s1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_6"/>
<font color="Maroon" title="c8">s1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:42_1_1_6"/><span class="lab"><font color="Green" title="E80">A40</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">s1</font>
 <span class="kw">and </span><br/><a NAME="E4:42_1_1_6"/><span class="lab"><font color="Green" title="E81">A41</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:42_1_1_6"/>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_6"><span class="lab"><font color="Green" title="E80">A40</font></span></a>, <a class="txt" href="#E4:42_1_1_6"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:42_1_1_6"/><span class="kw">then </span>
<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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:42_1_1_6"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:42_1_1"><span class="lab"><font color="Green" title="E78">A38</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:42_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A42</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42_1_1"><span class="lab"><font color="Green" title="E74">A34</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E9:42_1_1_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_6"><span class="lab"><font color="Green" title="E80">A40</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:42_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E8:42_1_1_6"><span class="lab"><font color="Green" title="E82">A42</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E40:42_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">a</font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42_1_1"><span class="lab"><font color="Green" title="E74">A34</font></span></a>, <a class="txt" href="#E28:42_1_1"><span class="lab"><font color="Green" title="E72">A32</font></span></a>, <a class="ref" href="topreal3.html#T38" title="TOPREAL3:th.38">TOPREAL3:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>
;<br/><a NAME="E42:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A43</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_8"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1_8"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">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="42_1_1_8_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_8_1"/><span class="lab"><font color="Green" title="E81">A44</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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">C</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">b<sub>2</sub></font> &amp; <font color="Olive" title="b1">C</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">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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="42_1_1_8_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_8_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1_8_1_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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="42_1_1_8_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_8_1_1_1_1"/><span class="lab"><font color="Green" title="E82">A45</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="TOPREAL1:func.3">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#K2" title="TOPREAL1:func.2">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/><span class="kw">reconsider </span><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">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="E4:42_1_1_8_1_1_1_1"/><span class="lab"><font color="Green" title="E83">A46</font></span>: 
( <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> &amp;  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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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">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="c7">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="c7">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E42:42_1_1"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E39:42_1_1"><span class="lab"><font color="Green" title="E79">A39</font></span></a>, <a class="txt" href="#E1:42_1_1_8_1"><span class="lab"><font color="Green" title="E81">A44</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:42_1_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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E1:42_1_1_8_1_1_1_1"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="ref" href="jordan12.html#T33" target="_self" title="JORDAN12:th.33">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_1_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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E26:42_1_1"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="txt" href="#E17:42_1_1"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="txt" href="#E3:42_1_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E32:42_1_1"><span class="lab"><font color="Green" title="E76">A36</font></span></a>, <a class="txt" href="#E4:42_1_1_8_1_1_1_1"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_8_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_8_1_1_1_2"/><span class="lab"><font color="Green" title="E82">A47</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="TOPREAL1:func.3">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#K2" title="TOPREAL1:func.2">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/><span class="kw">reconsider </span><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">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="E4:42_1_1_8_1_1_1_2"/><span class="lab"><font color="Green" title="E83">A48</font></span>: 
( <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> &amp;  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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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">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="c7">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="c7">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E42:42_1_1"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E39:42_1_1"><span class="lab"><font color="Green" title="E79">A39</font></span></a>, <a class="txt" href="#E1:42_1_1_8_1"><span class="lab"><font color="Green" title="E81">A44</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:42_1_1_8_1_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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E1:42_1_1_8_1_1_1_2"><span class="lab"><font color="Green" title="E82">A47</font></span></a>, <a class="ref" href="jordan12.html#T33" target="_self" title="JORDAN12:th.33">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_1_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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E26:42_1_1"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="txt" href="#E17:42_1_1"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="txt" href="#E3:42_1_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E32:42_1_1"><span class="lab"><font color="Green" title="E76">A36</font></span></a>, <a class="txt" href="#E4:42_1_1_8_1_1_1_2"><span class="lab"><font color="Green" title="E83">A48</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:42_1_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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_8_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_8_2"/><span class="lab"><font color="Green" title="E81">A49</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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">C</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">b<sub>2</sub></font> &amp; <font color="Olive" title="b1">C</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">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c7">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="42_1_1_8_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_8_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1_8_2_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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="42_1_1_8_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_8_2_1_1_1"/><span class="lab"><font color="Green" title="E82">A50</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="TOPREAL1:func.3">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#K2" title="TOPREAL1:func.2">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/><span class="kw">reconsider </span><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">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="E4:42_1_1_8_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A51</font></span>: 
( <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> &amp; (  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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c7">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="c7">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="c7">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E42:42_1_1"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E39:42_1_1"><span class="lab"><font color="Green" title="E79">A39</font></span></a>, <a class="txt" href="#E1:42_1_1_8_2"><span class="lab"><font color="Green" title="E81">A49</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:42_1_1_8_2_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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E1:42_1_1_8_2_1_1_1"><span class="lab"><font color="Green" title="E82">A50</font></span></a>, <a class="ref" href="jordan12.html#T33" target="_self" title="JORDAN12:th.33">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_1_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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E26:42_1_1"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="txt" href="#E17:42_1_1"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="txt" href="#E3:42_1_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E32:42_1_1"><span class="lab"><font color="Green" title="E76">A36</font></span></a>, <a class="txt" href="#E4:42_1_1_8_2_1_1_1"><span class="lab"><font color="Green" title="E83">A51</font></span></a>, <a class="ref" href="jordan12.html#T16" target="_self" title="JORDAN12:th.16">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_8_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_8_2_1_1_2"/><span class="lab"><font color="Green" title="E82">A52</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="TOPREAL1:func.3">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#K2" title="TOPREAL1:func.2">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/><span class="kw">reconsider </span><font color="Maroon" title="c15">c</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">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="E4:42_1_1_8_2_1_1_2"/><span class="lab"><font color="Green" title="E83">A53</font></span>: 
( <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> &amp; (  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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c7">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="c7">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="c7">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E42:42_1_1"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E39:42_1_1"><span class="lab"><font color="Green" title="E79">A39</font></span></a>, <a class="txt" href="#E1:42_1_1_8_2"><span class="lab"><font color="Green" title="E81">A49</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:42_1_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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:42_1_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E1:42_1_1_8_2_1_1_2"><span class="lab"><font color="Green" title="E82">A52</font></span></a>, <a class="ref" href="jordan12.html#T33" target="_self" title="JORDAN12:th.33">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_1_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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E26:42_1_1"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="txt" href="#E11:42_1_1"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E13:42_1_1"><span class="lab"><font color="Green" title="E61">A21</font></span></a>, <a class="txt" href="#E17:42_1_1"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="txt" href="#E3:42_1_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E32:42_1_1"><span class="lab"><font color="Green" title="E76">A36</font></span></a>, <a class="txt" href="#E4:42_1_1_8_2_1_1_2"><span class="lab"><font color="Green" title="E83">A53</font></span></a>, <a class="ref" href="jordan12.html#T17" target="_self" title="JORDAN12:th.17">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:42_1_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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="c11">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="c11">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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:42_1"/>
<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]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:42"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A54</font></span>: 
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T69" title="RELAT_1:th.69">RELAT_1:69</a></span>;<br/>
<a NAME="E13:42"/><span class="lab"><font color="Green" title="E50">A55</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:42"/><span class="lab"><font color="Green" title="E51">A56</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A57</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:42_2"/>
<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E2:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A58</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E51">A56</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E3:42_2"/><span class="lab"><font color="Green" title="E54">A59</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span> implies (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> ) ) )</span><br/><span class="kw">assume </span><a NAME="E4:42_2"/><span class="lab"><font color="Green" title="E55">A60</font></span>: 
<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> ) )</span><br/><span class="lab"><font color="Green" title="E56">A61</font></span>: <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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><a NAME="E6:42_2"/>
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E7:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A62</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/> <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c6">a</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="ref" href="topreal3.html#T38" title="TOPREAL3:th.38">TOPREAL3:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">g1</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>
<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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">g1</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="ref" href="topreal1.html#T22" title="TOPREAL1:th.22">TOPREAL1:22</a></span>
<br/>.= 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">g1</font>,1)

;<br/><span class="kw">hence </span><a NAME="E9:42_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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E7:42_2"><span class="lab"><font color="Green" title="E57">A62</font></span></a>, <a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="ref" href="jordan12.html#T33" target="_self" title="JORDAN12:th.33">Th34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A63</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
;<br/>
<a NAME="E18:42"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E53">A63</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E48">A8</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E19:42"/>
(  <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#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E49">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
