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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="glib_003.html#NM3" title="GLIB_003:NM.3">VGraph</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:179"/><i><font color="Green" title="E99">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="lexbfs.html#V21" title="LEXBFS:attr.21">with_property_L3</a>
 <b>and </b><br/><a NAME="E2:179"/><i><font color="Green" title="E100">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">V</font> be    <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E3:179"/><i><font color="Green" title="E101">A3</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>assume </b><a NAME="E4:179"/>
not <font color="Maroon" title="c2">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">k</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:179"/><i><font color="Green" title="E102">A4</font></i>: 
<font color="Maroon" title="c3">k</font> <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="c2">V</font>
 <b>and </b><br/><a NAME="E7:179"/><i><font color="Green" title="E103">A5</font></i>: 
 ex <font color="Olive" title="b1">H</font> being  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">k</font></span>)</span> ex <font color="Olive" title="b2">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">H</font> st <br/>( <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font> &amp; not <font color="Olive" title="b2">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="ref" href="chord.html#D13" title="CHORD:def.13">CHORD:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">HH</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">k</font></span>)</span>, <font color="Maroon" title="c5">hv</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">HH</font><b> such that </b><br/><a NAME="E9:179"/><i><font color="Green" title="E104">A6</font></i>: 
<font color="Maroon" title="c5">hv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
 <b>and </b><br/><a NAME="E10:179"/><i><font color="Green" title="E105">A7</font></i>: 
not <font color="Maroon" title="c5">hv</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E7:179"><i><font color="Green" title="E103">A5</font></i></a></i>;<br/>
<a NAME="E11:179"/><i><font color="Green" title="E106">A8</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">VLG</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E12:179"/><i><font color="Green" title="E107">A9</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="chord.html#D12" title="CHORD:def.12">CHORD:def 12</a></i>;<br/>
<a NAME="E13:179"/><b>then </b><i><font color="Green" title="E108">A10</font></i>: 
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E3:179"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="funct_1.html#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/>
<a NAME="E14:179"/><i><font color="Green" title="E109">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="chord.html#D12" title="CHORD:def.12">CHORD:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">ha</font>, <font color="Maroon" title="c8">hb</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">HH</font><b> such that </b><br/><a NAME="E16:179"/><i><font color="Green" title="E110">A12</font></i>: 
( <font color="Maroon" title="c7">ha</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">hb</font> &amp; <font color="Maroon" title="c5">hv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">ha</font> &amp; <font color="Maroon" title="c5">hv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">hb</font> )
 <b>and </b><br/><a NAME="E17:179"/><i><font color="Green" title="E111">A13</font></i>: 
<font color="Maroon" title="c5">hv</font>,<font color="Maroon" title="c7">ha</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E18:179"/><i><font color="Green" title="E112">A14</font></i>: 
<font color="Maroon" title="c5">hv</font>,<font color="Maroon" title="c8">hb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E19:179"/><i><font color="Green" title="E113">A15</font></i>: 
not <font color="Maroon" title="c7">ha</font>,<font color="Maroon" title="c8">hb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E10:179"><i><font color="Green" title="E105">A7</font></i></a>, <a class="ref" href="chord.html#T69" title="CHORD:th.69">CHORD:69</a></i>;<br/>
<a NAME="E20:179"/><i><font color="Green" title="E114">A16</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">k</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179"><i><font color="Green" title="E102">A4</font></i></a>, <a class="ref" href="chord.html#T107" title="CHORD:th.107">CHORD:107</a></i>;<br/>
<a NAME="E21:179"/><b>then </b><i><font color="Green" title="E115">A17</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">HH</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E22:179"/>
( <font color="Maroon" title="c5">hv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">HH</font> &amp; <font color="Maroon" title="c7">ha</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">HH</font> &amp; <font color="Maroon" title="c8">hb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">HH</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c5">hv</font>, <font color="Maroon" title="c10">aa</font> = <font color="Maroon" title="c7">ha</font>, <font color="Maroon" title="c11">bb</font> = <font color="Maroon" title="c8">hb</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E24:179"/><i><font color="Green" title="E116">A18</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/>( <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> iff <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:179_1"/><i><font color="Green" title="E117">A19</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> )
 <b>and </b><br/><a NAME="E2:179_1"/><i><font color="Green" title="E118">A20</font></i>: 
( <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> )
 ;<br/>

<a NAME="E3:179_1"/><i><font color="Green" title="E119">A21</font></i>: 
( <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E1:179_1"><i><font color="Green" title="E117">A19</font></i></a>, <a class="txt" href="lexbfs.html#E2:179_1"><i><font color="Green" title="E118">A20</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E4:179_1"/><b>then </b><i><font color="Green" title="E120">A22</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E3:179"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="lexbfs.html#E12:179"><i><font color="Green" title="E107">A9</font></i></a>, <a class="txt" href="lexbfs.html#E1:179_1"><i><font color="Green" title="E117">A19</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/>
<b>hence </b><a NAME="E5:179_1"/>
( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> implies <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">y</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:179"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="lexbfs.html#E12:179"><i><font color="Green" title="E107">A9</font></i></a>, <a class="txt" href="lexbfs.html#E1:179_1"><i><font color="Green" title="E117">A19</font></i></a>, <a class="txt" href="lexbfs.html#E3:179_1"><i><font color="Green" title="E119">A21</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/>

<b>thus </b><a NAME="E6:179_1"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">y</font> implies <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:179"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="lexbfs.html#E12:179"><i><font color="Green" title="E107">A9</font></i></a>, <a class="txt" href="lexbfs.html#E1:179_1"><i><font color="Green" title="E117">A19</font></i></a>, <a class="txt" href="lexbfs.html#E3:179_1"><i><font color="Green" title="E119">A21</font></i></a>, <a class="txt" href="lexbfs.html#E4:179_1"><i><font color="Green" title="E120">A22</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_2"><b>now </b></a><div class="add"><a NAME="E1:179_2"/><i><font color="Green" title="E117">A23</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">aa</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:179"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="lexbfs.html#E13:179"><i><font color="Green" title="E108">A10</font></i></a>, <a class="txt" href="lexbfs.html#E16:179"><i><font color="Green" title="E110">A12</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_2_1"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">aa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font> or <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">aa</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font> )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E1:179_2"><i><font color="Green" title="E117">A23</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_2_1_1"><b>suppose </b></a><a NAME="E1:179_2_1_1"/><i><font color="Green" title="E118">A24</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">aa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c12">aa</font> = <font color="Maroon" title="c10">aa</font>;<br/><b>take </b><font color="Maroon" title="c13">bb</font> = <font color="Maroon" title="c11">bb</font>;<br/><b>thus </b><a NAME="E2:179_2_1_1"/>
<font color="Maroon" title="c12">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E21:179"><i><font color="Green" title="E115">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E3:179_2_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">aa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">bb</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179_2_1_1"><i><font color="Green" title="E118">A24</font></i></a></i>;<br/><b>thus </b><a NAME="E4:179_2_1_1"/>
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c12">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E17:179"><i><font color="Green" title="E111">A13</font></i></a>, <a class="txt" href="lexbfs.html#E20:179"><i><font color="Green" title="E114">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E5:179_2_1_1"/>
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c13">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E18:179"><i><font color="Green" title="E112">A14</font></i></a>, <a class="txt" href="lexbfs.html#E20:179"><i><font color="Green" title="E114">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E6:179_2_1_1"/>
not <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E19:179"><i><font color="Green" title="E113">A15</font></i></a>, <a class="txt" href="lexbfs.html#E20:179"><i><font color="Green" title="E114">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_2_1_2"><b>suppose </b></a><a NAME="E1:179_2_1_2"/><i><font color="Green" title="E118">A25</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">aa</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c12">bb</font> = <font color="Maroon" title="c11">bb</font>;<br/><b>take </b><font color="Maroon" title="c13">aa</font> = <font color="Maroon" title="c10">aa</font>;<br/><b>thus </b><a NAME="E2:179_2_1_2"/>
<font color="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E21:179"><i><font color="Green" title="E115">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E3:179_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">aa</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">bb</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179_2_1_2"><i><font color="Green" title="E118">A25</font></i></a></i>;<br/><b>thus </b><a NAME="E4:179_2_1_2"/>
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c12">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E18:179"><i><font color="Green" title="E112">A14</font></i></a>, <a class="txt" href="lexbfs.html#E20:179"><i><font color="Green" title="E114">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E5:179_2_1_2"/>
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c13">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E17:179"><i><font color="Green" title="E111">A13</font></i></a>, <a class="txt" href="lexbfs.html#E20:179"><i><font color="Green" title="E114">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>;<br/><b>thus </b><a NAME="E6:179_2_1_2"/>
not <font color="Maroon" title="c12">bb</font>,<font color="Maroon" title="c13">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E19:179"><i><font color="Green" title="E113">A15</font></i></a>, <a class="txt" href="lexbfs.html#E20:179"><i><font color="Green" title="E114">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">bb</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E27:179"/><i><font color="Green" title="E117">A26</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">k</font>
 <b>and </b><br/><a NAME="E28:179"/><i><font color="Green" title="E118">A27</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">bb</font>
 <b>and </b><br/><a NAME="E29:179"/><i><font color="Green" title="E119">A28</font></i>: 
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c12">a</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E30:179"/><i><font color="Green" title="E120">A29</font></i>: 
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c13">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E31:179"/><i><font color="Green" title="E121">A30</font></i>: 
not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 &amp; <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  );<br/>
<a NAME="E32:179"/><i><font color="Green" title="E122">A31</font></i>: 
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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <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="c2">V</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:179"/><i><font color="Green" title="E123">A32</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c14">mbb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:179_3"/><i><font color="Green" title="E124">A33</font></i>: 
( <font color="Maroon" title="c14">mbb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c13">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">mbb</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E14:179"><i><font color="Green" title="E109">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">mbb</font> = <font color="Maroon" title="c14">mbb</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lexbfs.html#E2:179_3"><i><font color="Green" title="E124">A33</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">mbb</font>
;<br/>

<b>thus </b><a NAME="E4:179_3"/>
<font color="Maroon" title="c15">mbb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:179_3"><i><font color="Green" title="E124">A33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:179_3"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mbb</font></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E28:179"><i><font color="Green" title="E118">A27</font></i></a>, <a class="txt" href="lexbfs.html#E2:179_3"><i><font color="Green" title="E124">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E6:179_3"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mbb</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E28:179"><i><font color="Green" title="E118">A27</font></i></a>, <a class="txt" href="lexbfs.html#E2:179_3"><i><font color="Green" title="E124">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E7:179_3"/>
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mbb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E30:179"><i><font color="Green" title="E120">A29</font></i></a>, <a class="txt" href="lexbfs.html#E2:179_3"><i><font color="Green" title="E124">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E8:179_3"/>
not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mbb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E31:179"><i><font color="Green" title="E121">A30</font></i></a>, <a class="txt" href="lexbfs.html#E2:179_3"><i><font color="Green" title="E124">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">mb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E35:179"/><i><font color="Green" title="E124">A34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">mb</font>]
 <b>and </b><br/><a NAME="E36:179"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">mb</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="lexbfs.html#E32:179"><i><font color="Green" title="E122">A31</font></i></a>, <a class="txt" href="lexbfs.html#E33:179"><i><font color="Green" title="E123">A32</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">mb</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E38:179"/><i><font color="Green" title="E125">A35</font></i>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">b</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E29:179"><i><font color="Green" title="E119">A28</font></i></a>, <a class="txt" href="lexbfs.html#E35:179"><i><font color="Green" title="E124">A34</font></i></a></i>;<br/>
<a NAME="E39:179"/><i><font color="Green" title="E126">A36</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="lexbfs.html#E11:179"><i><font color="Green" title="E106">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E40:179"/><b>then </b><i><font color="Green" title="E127">A37</font></i>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E9:179"><i><font color="Green" title="E104">A6</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">ma</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E42:179"/><i><font color="Green" title="E128">A38</font></i>: 
( <font color="Maroon" title="c18">ma</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ma</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E14:179"><i><font color="Green" title="E109">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">ma</font> = <font color="Maroon" title="c18">ma</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lexbfs.html#E42:179"><i><font color="Green" title="E128">A38</font></i></a></i>;<br/>
<a NAME="E44:179"/><i><font color="Green" title="E129">A39</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">ma</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E42:179"><i><font color="Green" title="E128">A38</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E130">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:179_4"/>
<font color="Maroon" title="c19">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ;<br/><a NAME="E2:179_4"/><b>then </b><i><font color="Green" title="E131">A41</font></i>: 
<font color="Maroon" title="c19">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E9:179"><i><font color="Green" title="E104">A6</font></i></a>, <a class="txt" href="lexbfs.html#E30:179"><i><font color="Green" title="E120">A29</font></i></a>, <a class="txt" href="lexbfs.html#E31:179"><i><font color="Green" title="E121">A30</font></i></a>, <a class="txt" href="lexbfs.html#E42:179"><i><font color="Green" title="E128">A38</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:179_4"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E4:179_4"/><b>then </b>
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="chord.html#D12" title="CHORD:def.12">CHORD:def 12</a></i>;<br/><a NAME="E5:179_4"/><b>then </b>
<font color="Maroon" title="c16">a</font> <a href="chord.html#K3" title="CHORD:func.3">..</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E27:179"><i><font color="Green" title="E117">A26</font></i></a>, <a class="txt" href="lexbfs.html#E39:179"><i><font color="Green" title="E126">A36</font></i></a>, <a class="ref" href="chord.html#T16" title="CHORD:th.16">CHORD:16</a></i>;<br/><a NAME="E6:179_4"/><b>then </b>
<font color="Maroon" title="c16">a</font> <a href="chord.html#K3" title="CHORD:func.3">..</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">ma</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:179_4"><i><font color="Green" title="E131">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:179_4"/>
contradiction
 <b>by </b><i><a class="txt" href="lexbfs.html#E42:179"><i><font color="Green" title="E128">A38</font></i></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> ex <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font>, <font color="Olive" title="b4">v3</font>, <font color="Olive" title="b5">v4</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <font color="Olive" title="b1">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Olive" title="b1">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Olive" title="b4">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Olive" title="b5">v4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">v4</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">v3</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">v3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">v2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">v1</font> &amp; ( for <font color="Olive" title="b6">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">v4</font> ) &amp; ( for <font color="Olive" title="b6">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">v4</font> &amp; <font color="Olive" title="b6">x</font>,<font color="Olive" title="b3">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b6">x</font>,<font color="Olive" title="b2">v1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b6">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">v4</font> ) );<br/>
<a NAME="E46:179"/><i><font color="Green" title="E131">A42</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">v</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">a</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E24:179"><i><font color="Green" title="E116">A18</font></i></a>, <a class="txt" href="lexbfs.html#E39:179"><i><font color="Green" title="E126">A36</font></i></a>, <a class="txt" href="lexbfs.html#E40:179"><i><font color="Green" title="E127">A37</font></i></a>, <a class="txt" href="lexbfs.html#E42:179"><i><font color="Green" title="E128">A38</font></i></a>, <a class="txt" href="lexbfs.html#E44:179"><i><font color="Green" title="E129">A39</font></i></a>, <a class="txt" href="lexbfs.html#E45:179"><i><font color="Green" title="E130">A40</font></i></a></i>;<br/>
<a NAME="E47:179"/><i><font color="Green" title="E132">A43</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[4]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c20">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:179_5"/><i><font color="Green" title="E133">A44</font></i>: 
( <font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font> )
 <b>and </b><br/><a NAME="E3:179_5"/><i><font color="Green" title="E134">A45</font></i>: 
( <font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c16">a</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c15">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E4:179_5"/><i><font color="Green" title="E135">A46</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">c</font> &amp; <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c16">a</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c15">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="lexbfs.html#E2:179"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="lexbfs.html#E35:179"><i><font color="Green" title="E124">A34</font></i></a>, <a class="txt" href="lexbfs.html#E46:179"><i><font color="Green" title="E131">A42</font></i></a>, <a class="ref" href="lexbfs.html#D34" target="_self" title="LEXBFS:def.34">Def34</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">P</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c22">e1</font>, <font color="Maroon" title="c23">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:179_5"/><i><font color="Green" title="E136">A47</font></i>: 
( not <font color="Maroon" title="c21">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 &amp; <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>and </b><br/><a NAME="E7:179_5"/><i><font color="Green" title="E137">A48</font></i>: 
( <font color="Maroon" title="c22">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">b</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c23">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c22">e1</font>,<font color="Maroon" title="c23">e2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c17">b</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c16">a</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 <b>and </b><br/><a NAME="E8:179_5"/><i><font color="Green" title="E138">A49</font></i>: 
( <font color="Maroon" title="c21">P</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="c17">b</font> &amp; <font color="Maroon" title="c21">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c21">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E29:179"><i><font color="Green" title="E119">A28</font></i></a>, <a class="txt" href="lexbfs.html#E35:179"><i><font color="Green" title="E124">A34</font></i></a>, <a class="txt" href="lexbfs.html#E38:179"><i><font color="Green" title="E125">A35</font></i></a>, <a class="ref" href="chord.html#T47" title="CHORD:th.47">CHORD:47</a></i>;<br/>
<a NAME="E9:179_5"/>
( <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font> &amp; <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="txt" href="lexbfs.html#E8:179_5"><i><font color="Green" title="E138">A49</font></i></a></i>;<br/>
<a NAME="E10:179_5"/><b>then </b><i><font color="Green" title="E139">A50</font></i>: 
not <font color="Maroon" title="c21">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E35:179"><i><font color="Green" title="E124">A34</font></i></a>, <a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="ref" href="chord.html#T90" title="CHORD:th.90">CHORD:90</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:179_5"/><i><font color="Green" title="E140">A51</font></i>: 
<font color="Maroon" title="c24">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:179_5"><i><font color="Green" title="E134">A45</font></i></a>, <a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="txt" href="lexbfs.html#E8:179_5"><i><font color="Green" title="E138">A49</font></i></a>, <a class="ref" href="chord.html#D3" title="CHORD:def.3">CHORD:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">Qr</font> = <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font>;<br/>
<b>set </b><font color="Maroon" title="c26">Q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> ;<br/>
<a NAME="E13:179_5"/><i><font color="Green" title="E141">A52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="txt" href="lexbfs.html#E12:179_5"><i><font color="Green" title="E140">A51</font></i></a>, <a class="txt" href="lexbfs.html#E18"><i><font color="Green" title="E17">Lm4</font></i></a></i>;<br/>
<a NAME="E14:179_5"/><i><font color="Green" title="E142">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="txt" href="lexbfs.html#E12:179_5"><i><font color="Green" title="E140">A51</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/>
<a NAME="E15:179_5"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">P</font> &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">P</font> &amp; 5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">P</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E16:179_5"/><b>then </b><i><font color="Green" title="E143">A54</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <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="c17">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E8:179_5"><i><font color="Green" title="E138">A49</font></i></a>, <a class="txt" href="lexbfs.html#E12:179_5"><i><font color="Green" title="E140">A51</font></i></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></i>;<br/>
<a NAME="E17:179_5"/><i><font color="Green" title="E144">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E12:179_5"><i><font color="Green" title="E140">A51</font></i></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></i>;<br/>
<div><i><font color="Green" title="E145">A56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:179_5_1"/><i><font color="Green" title="E146">A57</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_5_1_1"/>
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font> or <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> or <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E7:179_5"><i><font color="Green" title="E137">A48</font></i></a>, <a class="txt" href="lexbfs.html#E1:179_5_1"><i><font color="Green" title="E146">A57</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1_1"><b>suppose </b></a><a NAME="E1:179_5_1_1_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1_2"><b>suppose </b></a><a NAME="E1:179_5_1_1_2"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E35:179"><i><font color="Green" title="E124">A34</font></i></a>, <a class="txt" href="lexbfs.html#E46:179"><i><font color="Green" title="E131">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_1_1_3"><b>suppose </b></a><a NAME="E1:179_5_1_1_3"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_1_1_3"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E35:179"><i><font color="Green" title="E124">A34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E19:179_5"/><b>then </b><i><font color="Green" title="E146">A58</font></i>: 
not <font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="lexbfs.html#E2:179_5"><i><font color="Green" title="E133">A44</font></i></a></i>;<br/>
<a NAME="E20:179_5"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c21">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E3:179_5"><i><font color="Green" title="E134">A45</font></i></a>, <a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="txt" href="lexbfs.html#E8:179_5"><i><font color="Green" title="E138">A49</font></i></a>, <a class="ref" href="chord.html#D3" title="CHORD:def.3">CHORD:def 3</a></i>;<br/>
<a NAME="E21:179_5"/><b>then </b>
( <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_5"><i><font color="Green" title="E136">A47</font></i></a>, <a class="txt" href="lexbfs.html#E10:179_5"><i><font color="Green" title="E139">A50</font></i></a>, <a class="txt" href="lexbfs.html#E12:179_5"><i><font color="Green" title="E140">A51</font></i></a>, <a class="txt" href="lexbfs.html#E19:179_5"><i><font color="Green" title="E146">A58</font></i></a>, <a class="ref" href="chord.html#T97" title="CHORD:th.97">CHORD:97</a></i>;<br/>
<a NAME="E22:179_5"/><b>then </b><i><font color="Green" title="E147">A59</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )
 
<b>by </b><i><a class="ref" href="chord.html#T91" title="CHORD:th.91">CHORD:91</a>, <a class="ref" href="glib_001.html#T121" title="GLIB_001:th.121">GLIB_001:121</a>, <a class="ref" href="glib_001.html#T149" title="GLIB_001:th.149">GLIB_001:149</a></i>;<br/>
<a NAME="E23:179_5"/><i><font color="Green" title="E148">A60</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> &amp; 5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> &amp; 7 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E14:179_5"><i><font color="Green" title="E142">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E24:179_5"/><i><font color="Green" title="E149">A61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E13:179_5"><i><font color="Green" title="E141">A52</font></i></a>, <a class="txt" href="lexbfs.html#E19"><i><font color="Green" title="E18">Lm5</font></i></a></i>;<br/>
<a NAME="E25:179_5"/><i><font color="Green" title="E150">A62</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="glib_001.html#T22" title="GLIB_001:th.22">GLIB_001:22</a></i>;<br/>
<a NAME="E26:179_5"/>
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 5</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 7</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E14:179_5"><i><font color="Green" title="E142">A53</font></i></a>, <a class="txt" href="lexbfs.html#E16:179_5"><i><font color="Green" title="E143">A54</font></i></a>, <a class="txt" href="lexbfs.html#E17:179_5"><i><font color="Green" title="E144">A55</font></i></a>, <a class="txt" href="lexbfs.html#E23:179_5"><i><font color="Green" title="E148">A60</font></i></a>, <a class="ref" href="glib_001.html#T25" title="GLIB_001:th.25">GLIB_001:25</a></i>;<br/>
<a NAME="E27:179_5"/><b>then </b><i><font color="Green" title="E151">A63</font></i>: 
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E14:179_5"><i><font color="Green" title="E142">A53</font></i></a>, <a class="txt" href="lexbfs.html#E25:179_5"><i><font color="Green" title="E150">A62</font></i></a></i>;<br/>
<a NAME="E28:179_5"/><i><font color="Green" title="E152">A64</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">a</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">v</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E24:179"><i><font color="Green" title="E116">A18</font></i></a>, <a class="txt" href="lexbfs.html#E35:179"><i><font color="Green" title="E124">A34</font></i></a>, <a class="txt" href="lexbfs.html#E39:179"><i><font color="Green" title="E126">A36</font></i></a>, <a class="txt" href="lexbfs.html#E40:179"><i><font color="Green" title="E127">A37</font></i></a>, <a class="txt" href="lexbfs.html#E42:179"><i><font color="Green" title="E128">A38</font></i></a>, <a class="txt" href="lexbfs.html#E44:179"><i><font color="Green" title="E129">A39</font></i></a>, <a class="txt" href="lexbfs.html#E45:179"><i><font color="Green" title="E130">A40</font></i></a>, <a class="txt" href="lexbfs.html#E2:179_5"><i><font color="Green" title="E133">A44</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:179_5_2"/><i><font color="Green" title="E153">A65</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><a NAME="E2:179_5_2"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c24">e</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179_5_2"><i><font color="Green" title="E153">A65</font></i></a>, <a class="ref" href="glib_001.html#T93" title="GLIB_001:th.93">GLIB_001:93</a></i>;<br/><a NAME="E3:179_5_2"/><b>then </b><i><font color="Green" title="E154">A66</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E12:179_5"><i><font color="Green" title="E140">A51</font></i></a>, <a class="ref" href="glib_001.html#T96" title="GLIB_001:th.96">GLIB_001:96</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_5_2_1"/>
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E3:179_5_2"><i><font color="Green" title="E154">A66</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_2_1_1"><b>suppose </b></a><a NAME="E1:179_5_2_1_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:179_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E18:179_5"><i><font color="Green" title="E145">A56</font></i></a></i>;<br/><b>hence </b><a NAME="E3:179_5_2_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:179_5"><i><font color="Green" title="E133">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_5_2_1_2"><b>suppose </b></a><a NAME="E1:179_5_2_1_2"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_5_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E30:179_5"/>
<font color="Maroon">S<sub>2</sub></font>[4]
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:179_5"><i><font color="Green" title="E135">A46</font></i></a>, <a class="txt" href="lexbfs.html#E22:179_5"><i><font color="Green" title="E147">A59</font></i></a>, <a class="txt" href="lexbfs.html#E24:179_5"><i><font color="Green" title="E149">A61</font></i></a>, <a class="txt" href="lexbfs.html#E27:179_5"><i><font color="Green" title="E151">A63</font></i></a>, <a class="txt" href="lexbfs.html#E28:179_5"><i><font color="Green" title="E152">A64</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:179"/><i><font color="Green" title="E133">A67</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">kk</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:179_6"/><i><font color="Green" title="E134">A68</font></i>: 
4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">kk</font>
 <b>and </b><br/><a NAME="E2:179_6"/><i><font color="Green" title="E135">A69</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">kk</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c21">k</font> = <font color="Maroon" title="c20">kk</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="lexbfs.html#E1:179_6"><i><font color="Green" title="E134">A68</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">P</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c23">v1</font>, <font color="Maroon" title="c24">v2</font>, <font color="Maroon" title="c25">v3</font>, <font color="Maroon" title="c26">v4</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:179_6"/><i><font color="Green" title="E136">A70</font></i>: 
( <font color="Maroon" title="c22">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <font color="Maroon" title="c22">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c22">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> &amp; <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <b>and </b><br/><a NAME="E6:179_6"/><i><font color="Green" title="E137">A71</font></i>: 
( <font color="Maroon" title="c23">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <font color="Maroon" title="c24">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Maroon" title="c25">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c26">v4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 <b>and </b><br/><a NAME="E7:179_6"/><i><font color="Green" title="E138">A72</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">v4</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">v3</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">v3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">v1</font> )
 <b>and </b><br/><a NAME="E8:179_6"/><i><font color="Green" title="E139">A73</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">v4</font>
 <b>and </b><br/><a NAME="E9:179_6"/><i><font color="Green" title="E140">A74</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">v4</font> &amp; <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c24">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c23">v1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">v4</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:179_6"><i><font color="Green" title="E135">A69</font></i></a></i>;<br/>
<a NAME="E10:179_6"/><i><font color="Green" title="E141">A75</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E5:179_6"><i><font color="Green" title="E136">A70</font></i></a>, <a class="ref" href="glib_001.html#T113" title="GLIB_001:th.113">GLIB_001:113</a></i>;<br/>
<a NAME="E11:179_6"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4
 
<b>by </b><i><a class="txt" href="lexbfs.html#E1:179_6"><i><font color="Green" title="E134">A68</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E12:179_6"/><b>then </b><i><font color="Green" title="E142">A76</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c21">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 8 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:179_6"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 3 <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="c22">P</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E10:179_6"><i><font color="Green" title="E141">A75</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c27">ez</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:179_6"/><i><font color="Green" title="E143">A77</font></i>: 
<font color="Maroon" title="c27">ez</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:179_6"><i><font color="Green" title="E136">A70</font></i></a>, <a class="ref" href="chord.html#T92" title="CHORD:th.92">CHORD:92</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c28">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:179_6_1"/><i><font color="Green" title="E144">A78</font></i>: 
<font color="Maroon" title="c28">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c26">v4</font>,<font color="Maroon" title="c25">v3</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E2:179_6_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c22">P</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E10:179_6"><i><font color="Green" title="E141">A75</font></i></a>, <a class="txt" href="lexbfs.html#E12:179_6"><i><font color="Green" title="E142">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:179_6_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:179_6"><i><font color="Green" title="E136">A70</font></i></a>, <a class="txt" href="lexbfs.html#E6:179_6"><i><font color="Green" title="E137">A71</font></i></a>, <a class="txt" href="lexbfs.html#E1:179_6_1"><i><font color="Green" title="E144">A78</font></i></a>, <a class="ref" href="chord.html#T92" title="CHORD:th.92">CHORD:92</a></i>;<br/><b>hence </b><a NAME="E4:179_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179_6"><i><font color="Green" title="E134">A68</font></i></a>, <a class="txt" href="lexbfs.html#E10:179_6"><i><font color="Green" title="E141">A75</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:179_6"/><b>then </b>
( <font color="Maroon" title="c26">v4</font>,<font color="Maroon" title="c24">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c26">v4</font>,<font color="Maroon" title="c25">v3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_6"><i><font color="Green" title="E137">A71</font></i></a>, <a class="txt" href="lexbfs.html#E15:179_6"><i><font color="Green" title="E143">A77</font></i></a>, <a class="ref" href="chord.html#D3" title="CHORD:def.3">CHORD:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c28">v5</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E19:179_6"/><i><font color="Green" title="E144">A79</font></i>: 
( <font color="Maroon" title="c28">v5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">v4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font> )
 <b>and </b><br/><a NAME="E20:179_6"/><i><font color="Green" title="E145">A80</font></i>: 
( <font color="Maroon" title="c28">v5</font>,<font color="Maroon" title="c25">v3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c28">v5</font>,<font color="Maroon" title="c24">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E21:179_6"/><i><font color="Green" title="E146">A81</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c28">v5</font> &amp; <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c25">v3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c24">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="lexbfs.html#E2:179"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="lexbfs.html#E7:179_6"><i><font color="Green" title="E138">A72</font></i></a>, <a class="ref" href="lexbfs.html#D34" target="_self" title="LEXBFS:def.34">Def34</a></i>;<br/>
<div><i><font color="Green" title="E147">A82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:179_6_2"/><i><font color="Green" title="E148">A83</font></i>: 
<font color="Maroon" title="c28">v5</font>,<font color="Maroon" title="c23">v1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ;<br/><a NAME="E2:179_6_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">v4</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E7:179_6"><i><font color="Green" title="E138">A72</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:179_6_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">v1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E7:179_6"><i><font color="Green" title="E138">A72</font></i></a>, <a class="txt" href="lexbfs.html#E19:179_6"><i><font color="Green" title="E144">A79</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c29">v6</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:179_6_2"/><i><font color="Green" title="E149">A84</font></i>: 
( <font color="Maroon" title="c29">v6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">v6</font> )
 <b>and </b><br/><a NAME="E6:179_6_2"/><i><font color="Green" title="E150">A85</font></i>: 
( <font color="Maroon" title="c29">v6</font>,<font color="Maroon" title="c24">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c29">v6</font>,<font color="Maroon" title="c23">v1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E7:179_6_2"/>
for <font color="Olive" title="b1">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">v6</font> &amp; <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c24">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c23">v1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">v6</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="lexbfs.html#E2:179"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="lexbfs.html#E20:179_6"><i><font color="Green" title="E145">A80</font></i></a>, <a class="txt" href="lexbfs.html#E1:179_6_2"><i><font color="Green" title="E148">A83</font></i></a>, <a class="ref" href="lexbfs.html#D34" target="_self" title="LEXBFS:def.34">Def34</a></i>;<br/><b>thus </b><a NAME="E8:179_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="lexbfs.html#E9:179_6"><i><font color="Green" title="E140">A74</font></i></a>, <a class="txt" href="lexbfs.html#E19:179_6"><i><font color="Green" title="E144">A79</font></i></a>, <a class="txt" href="lexbfs.html#E5:179_6_2"><i><font color="Green" title="E149">A84</font></i></a>, <a class="txt" href="lexbfs.html#E6:179_6_2"><i><font color="Green" title="E150">A85</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c29">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:179_6"/><i><font color="Green" title="E148">A86</font></i>: 
<font color="Maroon" title="c29">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c28">v5</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E6:179_6"><i><font color="Green" title="E137">A71</font></i></a>, <a class="txt" href="lexbfs.html#E20:179_6"><i><font color="Green" title="E145">A80</font></i></a>, <a class="ref" href="chord.html#D3" title="CHORD:def.3">CHORD:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c30">Qr</font> = <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font>;<br/>
<b>set </b><font color="Maroon" title="c31">Q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> ;<br/>
<a NAME="E25:179_6"/><i><font color="Green" title="E149">A87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E5:179_6"><i><font color="Green" title="E136">A70</font></i></a>, <a class="txt" href="lexbfs.html#E24:179_6"><i><font color="Green" title="E148">A86</font></i></a>, <a class="txt" href="lexbfs.html#E18"><i><font color="Green" title="E17">Lm4</font></i></a></i>;<br/>
<a NAME="E26:179_6"/><i><font color="Green" title="E150">A88</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="c22">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="txt" href="lexbfs.html#E24:179_6"><i><font color="Green" title="E148">A86</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/>
<a NAME="E27:179_6"/>
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="c22">P</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E10:179_6"><i><font color="Green" title="E141">A75</font></i></a>, <a class="txt" href="lexbfs.html#E12:179_6"><i><font color="Green" title="E142">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E28:179_6"/><b>then </b>
( <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v3</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">P</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_6"><i><font color="Green" title="E137">A71</font></i></a>, <a class="txt" href="lexbfs.html#E26:179_6"><i><font color="Green" title="E150">A88</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E29:179_6"/><b>then </b><i><font color="Green" title="E151">A89</font></i>: 
( <font color="Maroon" title="c25">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E24:179_6"><i><font color="Green" title="E148">A86</font></i></a>, <a class="txt" href="lexbfs.html#E26:179_6"><i><font color="Green" title="E150">A88</font></i></a>, <a class="txt" href="lexbfs.html#E20"><i><font color="Green" title="E19">Lm6</font></i></a></i>;<br/>
<a NAME="E30:179_6"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 &amp; 3 <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="c22">P</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E10:179_6"><i><font color="Green" title="E141">A75</font></i></a>, <a class="txt" href="lexbfs.html#E12:179_6"><i><font color="Green" title="E142">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E31:179_6"/><b>then </b>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">P</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E32:179_6"/><b>then </b><i><font color="Green" title="E152">A90</font></i>: 
( <font color="Maroon" title="c24">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_6"><i><font color="Green" title="E137">A71</font></i></a>, <a class="txt" href="lexbfs.html#E24:179_6"><i><font color="Green" title="E148">A86</font></i></a>, <a class="txt" href="lexbfs.html#E20"><i><font color="Green" title="E19">Lm6</font></i></a></i>;<br/>
<a NAME="E33:179_6"/><i><font color="Green" title="E153">A91</font></i>: 
( <font color="Maroon" title="c26">v4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <font color="Maroon" title="c28">v5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_6"><i><font color="Green" title="E137">A71</font></i></a>, <a class="txt" href="lexbfs.html#E24:179_6"><i><font color="Green" title="E148">A86</font></i></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></i>;<br/>
<a NAME="E34:179_6"/><i><font color="Green" title="E154">A92</font></i>: 
not <font color="Maroon" title="c28">v5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="lexbfs.html#E8:179_6"><i><font color="Green" title="E139">A73</font></i></a>, <a class="txt" href="lexbfs.html#E19:179_6"><i><font color="Green" title="E144">A79</font></i></a></i>;<br/>
<a NAME="E35:179_6"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Maroon" title="c28">v5</font>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:179_6"><i><font color="Green" title="E137">A71</font></i></a>, <a class="txt" href="lexbfs.html#E22:179_6"><i><font color="Green" title="E147">A82</font></i></a>, <a class="ref" href="chord.html#D3" title="CHORD:def.3">CHORD:def 3</a></i>;<br/>
<a NAME="E36:179_6"/><b>then </b>
( <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E5:179_6"><i><font color="Green" title="E136">A70</font></i></a>, <a class="txt" href="lexbfs.html#E24:179_6"><i><font color="Green" title="E148">A86</font></i></a>, <a class="txt" href="lexbfs.html#E34:179_6"><i><font color="Green" title="E154">A92</font></i></a>, <a class="ref" href="chord.html#T97" title="CHORD:th.97">CHORD:97</a></i>;<br/>
<a NAME="E37:179_6"/><b>then </b><i><font color="Green" title="E155">A93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )
 
<b>by </b><i><a class="ref" href="chord.html#T91" title="CHORD:th.91">CHORD:91</a>, <a class="ref" href="glib_001.html#T121" title="GLIB_001:th.121">GLIB_001:121</a>, <a class="ref" href="glib_001.html#T149" title="GLIB_001:th.149">GLIB_001:149</a></i>;<br/>
<a NAME="E38:179_6"/><i><font color="Green" title="E156">A94</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E25:179_6"><i><font color="Green" title="E149">A87</font></i></a>, <a class="txt" href="lexbfs.html#E19"><i><font color="Green" title="E18">Lm5</font></i></a></i>;<br/>
<a NAME="E39:179_6"/><i><font color="Green" title="E157">A95</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#T22" title="GLIB_001:th.22">GLIB_001:22</a></i>;<br/>
<a NAME="E40:179_6"/><b>then </b>
( <font color="Maroon" title="c24">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c25">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E29:179_6"><i><font color="Green" title="E151">A89</font></i></a>, <a class="txt" href="lexbfs.html#E32:179_6"><i><font color="Green" title="E152">A90</font></i></a>, <a class="ref" href="glib_001.html#T25" title="GLIB_001:th.25">GLIB_001:25</a></i>;<br/>
<a NAME="E41:179_6"/><b>then </b><i><font color="Green" title="E158">A96</font></i>: 
( <font color="Maroon" title="c24">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <font color="Maroon" title="c25">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Maroon" title="c26">v4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c28">v5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E33:179_6"><i><font color="Green" title="E153">A91</font></i></a>, <a class="txt" href="lexbfs.html#E39:179_6"><i><font color="Green" title="E157">A95</font></i></a>, <a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:179_6_3"/><i><font color="Green" title="E159">A97</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><a NAME="E2:179_6_3"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c29">e</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:179_6_3"><i><font color="Green" title="E159">A97</font></i></a>, <a class="ref" href="glib_001.html#T93" title="GLIB_001:th.93">GLIB_001:93</a></i>;<br/><a NAME="E3:179_6_3"/><b>then </b><i><font color="Green" title="E160">A98</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c28">v5</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E24:179_6"><i><font color="Green" title="E148">A86</font></i></a>, <a class="ref" href="glib_001.html#T96" title="GLIB_001:th.96">GLIB_001:96</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_6_3_1"/>
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c28">v5</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E3:179_6_3"><i><font color="Green" title="E160">A98</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_6_3_1_1"><b>suppose </b></a><a NAME="E1:179_6_3_1_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:179_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">v4</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E8:179_6"><i><font color="Green" title="E139">A73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:179_6_3_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E19:179_6"><i><font color="Green" title="E144">A79</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_6_3_1_2"><b>suppose </b></a><a NAME="E1:179_6_3_1_2"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c28">v5</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_6_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E43:179_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="lexbfs.html#E7:179_6"><i><font color="Green" title="E138">A72</font></i></a>, <a class="txt" href="lexbfs.html#E19:179_6"><i><font color="Green" title="E144">A79</font></i></a>, <a class="txt" href="lexbfs.html#E21:179_6"><i><font color="Green" title="E146">A81</font></i></a>, <a class="txt" href="lexbfs.html#E37:179_6"><i><font color="Green" title="E155">A93</font></i></a>, <a class="txt" href="lexbfs.html#E38:179_6"><i><font color="Green" title="E156">A94</font></i></a>, <a class="txt" href="lexbfs.html#E41:179_6"><i><font color="Green" title="E158">A96</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:179"/><i><font color="Green" title="E134">A99</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="lexbfs.html#E47:179"><i><font color="Green" title="E132">A43</font></i></a>, <a class="txt" href="lexbfs.html#E48:179"><i><font color="Green" title="E133">A67</font></i></a>);<br/></i>
<a NAME="E50:179"/>
( 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 11 &amp; 11 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 11 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E51:179"/><b>then </b>
4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 11
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E52:179"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 11]
 
<b>by </b><i><a class="txt" href="lexbfs.html#E49:179"><i><font color="Green" title="E134">A99</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">P</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c21">v1</font>, <font color="Maroon" title="c22">v2</font>, <font color="Maroon" title="c23">v3</font>, <font color="Maroon" title="c24">v4</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E54:179"/><i><font color="Green" title="E135">A100</font></i>: 
( <font color="Maroon" title="c20">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <font color="Maroon" title="c20">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c20">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )
 <b>and </b><br/><a NAME="E55:179"/><i><font color="Green" title="E136">A101</font></i>: 
<font color="Maroon" title="c20">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 11</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>and </b><br/><a NAME="E56:179"/>
( <font color="Maroon" title="c21">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <font color="Maroon" title="c22">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Maroon" title="c23">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c24">v4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 <b>and </b><br/><a NAME="E57:179"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v4</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">v3</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">v3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">v2</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">v2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">v1</font> )
 <b>and </b><br/><a NAME="E58:179"/>
for <font color="Olive" title="b1">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">v4</font> &amp; <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c22">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c21">v1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v4</font>
 ;<br/>
<a NAME="E59:179"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 10</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E55:179"><i><font color="Green" title="E136">A101</font></i></a>, <a class="ref" href="glib_001.html#T113" title="GLIB_001:th.113">GLIB_001:113</a></i>;<br/>
<a NAME="E60:179"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 21</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">P</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D14" title="GLIB_001:def.14">GLIB_001:def 14</a></i>;<br/>
<a NAME="E61:179"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 11 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E54:179"><i><font color="Green" title="E135">A100</font></i></a>, <a class="ref" href="glib_001.html#T155" title="GLIB_001:th.155">GLIB_001:155</a></i>;<br/>
<b>hence </b><a NAME="E62:179"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>


</div>
