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<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="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">L</font> is <a href="lexbfs.html#V6" title="LEXBFS:attr.6">with_property_L3</a> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">L</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> holds <br/>for <font color="Olive" title="b2">V</font> being   <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">V</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> holds <br/><font color="Olive" title="b2">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is <a href="lexbfs.html#V6" title="LEXBFS:attr.6">with_property_L3</a> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">L</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> implies for <font color="Olive" title="b1">V</font> being   <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">V</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> holds <br/><font color="Olive" title="b1">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:103"/><i><font color="Green" title="E78">A1</font></i>: 
<font color="Maroon" title="c2">L</font> is <a href="lexbfs.html#V6" title="LEXBFS:attr.6">with_property_L3</a>
 <b>and </b><br/><a NAME="E2:103"/><i><font color="Green" title="E79">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">L</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being   <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">V</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> holds <br/><font color="Olive" title="b1">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a></span><br/>

<b>let </b><font color="Maroon" title="c3">V</font> be    <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">V</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> implies <font color="Maroon" title="c3">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> )</span><br/>

<b>assume </b><a NAME="E3:103"/><i><font color="Green" title="E80">A3</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a></span><br/>

<b>assume </b><a NAME="E4:103"/>
not <font color="Maroon" title="c3">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c4">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="E6:103"/><i><font color="Green" title="E81">A4</font></i>: 
<font color="Maroon" title="c4">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="c3">V</font>
 <b>and </b><br/><a NAME="E7:103"/><i><font color="Green" title="E82">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="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">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="c3">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c5">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="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font></span>)</span>, <font color="Maroon" title="c6">hv</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">HH</font><b> such that </b><br/><a NAME="E9:103"/><i><font color="Green" title="E83">A6</font></i>: 
<font color="Maroon" title="c6">hv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <b>and </b><br/><a NAME="E10:103"/><i><font color="Green" title="E84">A7</font></i>: 
not <font color="Maroon" title="c6">hv</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 <b>by </b><i><a class="txt" href="#E7:103"><i><font color="Green" title="E82">A5</font></i></a></i>;<br/>
<a NAME="E11:103"/><i><font color="Green" title="E85">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="c4">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/>
<a NAME="E12:103"/><i><font color="Green" title="E86">A9</font></i>: 
<font color="Maroon" title="c3">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="E13:103"/><i><font color="Green" title="E87">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">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="c5">HH</font><b> such that </b><br/><a NAME="E15:103"/><i><font color="Green" title="E88">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="c6">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="c6">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="E16:103"/><i><font color="Green" title="E89">A13</font></i>: 
<font color="Maroon" title="c6">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="E17:103"/><i><font color="Green" title="E90">A14</font></i>: 
<font color="Maroon" title="c6">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="E18:103"/><i><font color="Green" title="E91">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="#E10:103"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="chord.html#T69" title="CHORD:th.69">CHORD:69</a></i>;<br/>
<a NAME="E19:103"/><i><font color="Green" title="E92">A16</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">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="#E6:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="chord.html#T107" title="CHORD:th.107">CHORD:107</a></i>;<br/>
<a NAME="E20:103"/><b>then </b><i><font color="Green" title="E93">A17</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c5">HH</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">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="E21:103"/>
( <font color="Maroon" title="c6">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="c5">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="c5">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="c5">HH</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c6">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="E23:103"/><i><font color="Green" title="E94">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="c3">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="c3">V</font> &amp; <font color="Maroon" title="c3">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="c3">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 <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="103_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c3">V</font> &amp; <font color="Olive" title="b2">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="c3">V</font> &amp; <font color="Maroon" title="c3">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c3">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> holds <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> iff <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">y</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c3">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="c3">V</font> &amp; <font color="Maroon" title="c3">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="c3">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> 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> iff <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">y</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:103_1"/><i><font color="Green" title="E95">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="c3">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="c3">V</font> )
 <b>and </b><br/><a NAME="E2:103_1"/><i><font color="Green" title="E96">A20</font></i>: 
( <font color="Maroon" title="c3">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="c3">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> iff <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">y</font> )</span><br/>

<a NAME="E3:103_1"/><i><font color="Green" title="E97">A21</font></i>: 
( <font color="Maroon" title="c3">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="c3">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="#E1:103_1"><i><font color="Green" title="E95">A19</font></i></a>, <a class="txt" href="#E2:103_1"><i><font color="Green" title="E96">A20</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E4:103_1"/><b>then </b><i><font color="Green" title="E98">A22</font></i>: 
<font color="Maroon" title="c2">L</font> <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="#E3:103"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="#E12:103"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="#E1:103_1"><i><font color="Green" title="E95">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="E5:103_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 <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E3:103"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="#E12:103"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="#E1:103_1"><i><font color="Green" title="E95">A19</font></i></a>, <a class="txt" href="#E3:103_1"><i><font color="Green" title="E97">A21</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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> )</span><br/>

<b>thus </b><a NAME="E6:103_1"/>
( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E3:103"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="#E12:103"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="#E1:103_1"><i><font color="Green" title="E95">A19</font></i></a>, <a class="txt" href="#E3:103_1"><i><font color="Green" title="E97">A21</font></i></a>, <a class="txt" href="#E4:103_1"><i><font color="Green" title="E98">A22</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2"><b>now </b></a><div class="add"><a NAME="E1:103_2"/><i><font color="Green" title="E95">A23</font></i>: 
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E2:103"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="#E12:103"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="#E3:103"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="#E15:103"><i><font color="Green" title="E88">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="103_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_2_1"/>
( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font> or <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E1:103_2"><i><font color="Green" title="E95">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_2_1_1"><b>suppose </b></a><a NAME="E1:103_2_1_1"/><i><font color="Green" title="E96">A24</font></i>: 
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">aa</font>, <font color="Olive" title="b2">bb</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">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">aa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">bb</font> &amp; <font color="Maroon" title="c9">v</font>,<font color="Olive" title="b1">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c9">v</font>,<font color="Olive" title="b2">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b1">aa</font>,<font color="Olive" title="b2">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c12">aa</font> = <font color="Maroon" title="c10">aa</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">bb</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="Maroon" title="c12">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">bb</font> &amp; <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>  &amp; <font color="Maroon" title="c9">v</font>,<font color="Olive" title="b1">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c12">aa</font>,<font color="Olive" title="b1">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><b>take </b><font color="Maroon" title="c13">bb</font> = <font color="Maroon" title="c11">bb</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">bb</font> &amp; <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>  &amp; <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E2:103_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="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="#E20:103"><i><font color="Green" title="E93">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">bb</font> &amp; <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>  &amp; <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E3:103_2_1_1"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E1:103_2_1_1"><i><font color="Green" title="E96">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  &amp; <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E4:103_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="#E16:103"><i><font color="Green" title="E89">A13</font></i></a>, <a class="txt" href="#E19:103"><i><font color="Green" title="E92">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E5:103_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="#E17:103"><i><font color="Green" title="E90">A14</font></i></a>, <a class="txt" href="#E19:103"><i><font color="Green" title="E92">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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> </span><br/><b>thus </b><a NAME="E6:103_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="#E18:103"><i><font color="Green" title="E91">A15</font></i></a>, <a class="txt" href="#E19:103"><i><font color="Green" title="E92">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_2_1_2"><b>suppose </b></a><a NAME="E1:103_2_1_2"/><i><font color="Green" title="E96">A25</font></i>: 
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">bb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">bb</font>, <font color="Olive" title="b2">aa</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">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">aa</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">bb</font> &amp; <font color="Maroon" title="c9">v</font>,<font color="Olive" title="b1">bb</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c9">v</font>,<font color="Olive" title="b2">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Olive" title="b1">bb</font>,<font color="Olive" title="b2">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c12">bb</font> = <font color="Maroon" title="c11">bb</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">aa</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="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">aa</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">bb</font> &amp; <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>  &amp; <font color="Maroon" title="c9">v</font>,<font color="Olive" title="b1">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c12">bb</font>,<font color="Olive" title="b1">aa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><b>take </b><font color="Maroon" title="c13">aa</font> = <font color="Maroon" title="c10">aa</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">bb</font> &amp; <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>  &amp; <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E2:103_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="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="#E20:103"><i><font color="Green" title="E93">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">bb</font> &amp; <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>  &amp; <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E3:103_2_1_2"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E1:103_2_1_2"><i><font color="Green" title="E96">A25</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  &amp; <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E4:103_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="#E17:103"><i><font color="Green" title="E90">A14</font></i></a>, <a class="txt" href="#E19:103"><i><font color="Green" title="E92">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  &amp; 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>  )</span><br/><b>thus </b><a NAME="E5:103_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="#E16:103"><i><font color="Green" title="E89">A13</font></i></a>, <a class="txt" href="#E19:103"><i><font color="Green" title="E92">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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> </span><br/><b>thus </b><a NAME="E6:103_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="#E18:103"><i><font color="Green" title="E91">A15</font></i></a>, <a class="txt" href="#E19:103"><i><font color="Green" title="E92">A16</font></i></a>, <a class="ref" href="chord.html#T45" title="CHORD:th.45">CHORD:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="E26:103"/><i><font color="Green" title="E95">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="c3">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c4">k</font>
 <b>and </b><br/><a NAME="E27:103"/><i><font color="Green" title="E96">A27</font></i>: 
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="E28:103"/><i><font color="Green" title="E97">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="E29:103"/><i><font color="Green" title="E98">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="E30:103"/><i><font color="Green" title="E99">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="c3">V</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c3">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="c3">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="c3">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="E31:103"/><i><font color="Green" title="E100">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="c3">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="E32:103"/><i><font color="Green" title="E101">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="103_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="E2:103_3"/><i><font color="Green" title="E102">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="c3">V</font> &amp; <font color="Maroon" title="c13">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="#E13:103"><i><font color="Green" title="E87">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="#E2:103_3"><i><font color="Green" title="E102">A33</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">mbb</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">mbb</font>]</span><br/>

<b>thus </b><a NAME="E4:103_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="c3">V</font>
 <b>by </b><i><a class="txt" href="#E2:103_3"><i><font color="Green" title="E102">A33</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mbb</font></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="c3">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mbb</font> &amp; <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c3">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>  &amp; not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">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>  )</span><br/>

<b>thus </b><a NAME="E5:103_3"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="#E27:103"><i><font color="Green" title="E96">A27</font></i></a>, <a class="txt" href="#E2:103_3"><i><font color="Green" title="E102">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mbb</font> &amp; <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c3">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>  &amp; not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">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>  )</span><br/>

<b>thus </b><a NAME="E6:103_3"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">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="#E27:103"><i><font color="Green" title="E96">A27</font></i></a>, <a class="txt" href="#E2:103_3"><i><font color="Green" title="E102">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c3">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>  &amp; not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">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>  )</span><br/>

<b>thus </b><a NAME="E7:103_3"/>
<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c3">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="#E29:103"><i><font color="Green" title="E98">A29</font></i></a>, <a class="txt" href="#E2:103_3"><i><font color="Green" title="E102">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">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> </span><br/>

<b>thus </b><a NAME="E8:103_3"/>
not <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">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="#E30:103"><i><font color="Green" title="E99">A30</font></i></a>, <a class="txt" href="#E2:103_3"><i><font color="Green" title="E102">A33</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="E34:103"/><i><font color="Green" title="E102">A34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">mb</font>]
 <b>and </b><br/><a NAME="E35:103"/>
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="#E31:103"><i><font color="Green" title="E100">A31</font></i></a>, <a class="txt" href="#E32:103"><i><font color="Green" title="E101">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="c3">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="E37:103"/><i><font color="Green" title="E103">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="#E28:103"><i><font color="Green" title="E97">A28</font></i></a>, <a class="txt" href="#E34:103"><i><font color="Green" title="E102">A34</font></i></a></i>;<br/>
<a NAME="E38:103"/><i><font color="Green" title="E104">A36</font></i>: 
<font color="Maroon" title="c4">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="c3">V</font>
 
<b>by </b><i><a class="txt" href="#E6:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="#E11:103"><i><font color="Green" title="E85">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="E39:103"/><b>then </b><i><font color="Green" title="E105">A37</font></i>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="txt" href="#E9:103"><i><font color="Green" title="E83">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="E41:103"/><i><font color="Green" title="E106">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="c3">V</font> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="#E13:103"><i><font color="Green" title="E87">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="#E41:103"><i><font color="Green" title="E106">A38</font></i></a></i>;<br/>
<a NAME="E43:103"/><i><font color="Green" title="E107">A39</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="#E41:103"><i><font color="Green" title="E106">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="E108">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_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="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:103_4"/><b>then </b><i><font color="Green" title="E109">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="c4">k</font>
 <b>by </b><i><a class="txt" href="#E9:103"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="#E29:103"><i><font color="Green" title="E98">A29</font></i></a>, <a class="txt" href="#E30:103"><i><font color="Green" title="E99">A30</font></i></a>, <a class="txt" href="#E41:103"><i><font color="Green" title="E106">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:103_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="E4:103_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="c3">V</font> &amp; <font color="Maroon" title="c3">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="E5:103_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="c3">V</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="#E26:103"><i><font color="Green" title="E95">A26</font></i></a>, <a class="txt" href="#E38:103"><i><font color="Green" title="E104">A36</font></i></a>, <a class="ref" href="chord.html#T16" title="CHORD:th.16">CHORD:16</a></i>;<br/><a NAME="E6:103_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="c3">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="#E2:103_4"><i><font color="Green" title="E109">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="E7:103_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E41:103"><i><font color="Green" title="E106">A38</font></i></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">v3</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">v4</font> ) );<br/>
<a NAME="E45:103"/><i><font color="Green" title="E109">A42</font></i>: 
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E23:103"><i><font color="Green" title="E94">A18</font></i></a>, <a class="txt" href="#E38:103"><i><font color="Green" title="E104">A36</font></i></a>, <a class="txt" href="#E39:103"><i><font color="Green" title="E105">A37</font></i></a>, <a class="txt" href="#E41:103"><i><font color="Green" title="E106">A38</font></i></a>, <a class="txt" href="#E43:103"><i><font color="Green" title="E107">A39</font></i></a>, <a class="txt" href="#E44:103"><i><font color="Green" title="E108">A40</font></i></a></i>;<br/>
<a NAME="E46:103"/><i><font color="Green" title="E110">A43</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[4]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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="E2:103_5"/><i><font color="Green" title="E111">A44</font></i>: 
( <font color="Maroon" title="c20">c</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">L</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="E3:103_5"/><i><font color="Green" title="E112">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="E4:103_5"/><i><font color="Green" title="E113">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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="#E2:103"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="#E34:103"><i><font color="Green" title="E102">A34</font></i></a>, <a class="txt" href="#E45:103"><i><font color="Green" title="E109">A42</font></i></a>, <a class="ref" href="lexbfs.html#D18" target="_self" title="LEXBFS:def.18">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="E6:103_5"/><i><font color="Green" title="E114">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="E7:103_5"/><i><font color="Green" title="E115">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="E8:103_5"/><i><font color="Green" title="E116">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="#E28:103"><i><font color="Green" title="E97">A28</font></i></a>, <a class="txt" href="#E34:103"><i><font color="Green" title="E102">A34</font></i></a>, <a class="txt" href="#E37:103"><i><font color="Green" title="E103">A35</font></i></a>, <a class="ref" href="chord.html#T47" title="CHORD:th.47">CHORD:47</a></i>;<br/>
<a NAME="E9:103_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="#E6:103_5"><i><font color="Green" title="E114">A47</font></i></a>, <a class="txt" href="#E8:103_5"><i><font color="Green" title="E116">A49</font></i></a></i>;<br/>
<a NAME="E10:103_5"/><b>then </b><i><font color="Green" title="E117">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="#E34:103"><i><font color="Green" title="E102">A34</font></i></a>, <a class="txt" href="#E6:103_5"><i><font color="Green" title="E114">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="E12:103_5"/><i><font color="Green" title="E118">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="#E3:103_5"><i><font color="Green" title="E112">A45</font></i></a>, <a class="txt" href="#E6:103_5"><i><font color="Green" title="E114">A47</font></i></a>, <a class="txt" href="#E8:103_5"><i><font color="Green" title="E116">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="E13:103_5"/><i><font color="Green" title="E119">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="#E6:103_5"><i><font color="Green" title="E114">A47</font></i></a>, <a class="txt" href="#E12:103_5"><i><font color="Green" title="E118">A51</font></i></a>, <a class="txt" href="lexbfs.html#E17"><i><font color="Green" title="E16">Lm4</font></i></a></i>;<br/>
<a NAME="E14:103_5"/><i><font color="Green" title="E120">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="#E6:103_5"><i><font color="Green" title="E114">A47</font></i></a>, <a class="txt" href="#E12:103_5"><i><font color="Green" title="E118">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="E15:103_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="#E6:103_5"><i><font color="Green" title="E114">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="E16:103_5"/><b>then </b><i><font color="Green" title="E121">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="#E8:103_5"><i><font color="Green" title="E116">A49</font></i></a>, <a class="txt" href="#E12:103_5"><i><font color="Green" title="E118">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="E17:103_5"/><i><font color="Green" title="E122">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="#E12:103_5"><i><font color="Green" title="E118">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="E123">A56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  implies <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font> )</span><br/><b>assume </b><a NAME="E1:103_5_1"/><i><font color="Green" title="E124">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_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="#E7:103_5"><i><font color="Green" title="E115">A48</font></i></a>, <a class="txt" href="#E1:103_5_1"><i><font color="Green" title="E124">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="103_5_1_1_1"><b>suppose </b></a><a NAME="E1:103_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font></span><br/><div class="add"><b>hence </b><a NAME="E2:103_5_1_1_1"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_1_2"><b>suppose </b></a><a NAME="E1:103_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font></span><br/><div class="add"><b>hence </b><a NAME="E2:103_5_1_1_2"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E34:103"><i><font color="Green" title="E102">A34</font></i></a>, <a class="txt" href="#E45:103"><i><font color="Green" title="E109">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_1_3"><b>suppose </b></a><a NAME="E1:103_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font></span><br/><div class="add"><b>hence </b><a NAME="E2:103_5_1_1_3"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E34:103"><i><font color="Green" title="E102">A34</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E19:103_5"/><b>then </b><i><font color="Green" title="E124">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="#E2:103_5"><i><font color="Green" title="E111">A44</font></i></a></i>;<br/>
<a NAME="E20:103_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="#E3:103_5"><i><font color="Green" title="E112">A45</font></i></a>, <a class="txt" href="#E6:103_5"><i><font color="Green" title="E114">A47</font></i></a>, <a class="txt" href="#E8:103_5"><i><font color="Green" title="E116">A49</font></i></a>, <a class="ref" href="chord.html#D3" title="CHORD:def.3">CHORD:def 3</a></i>;<br/>
<a NAME="E21:103_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="#E6:103_5"><i><font color="Green" title="E114">A47</font></i></a>, <a class="txt" href="#E10:103_5"><i><font color="Green" title="E117">A50</font></i></a>, <a class="txt" href="#E12:103_5"><i><font color="Green" title="E118">A51</font></i></a>, <a class="txt" href="#E19:103_5"><i><font color="Green" title="E124">A58</font></i></a>, <a class="ref" href="chord.html#T97" title="CHORD:th.97">CHORD:97</a></i>;<br/>
<a NAME="E22:103_5"/><b>then </b><i><font color="Green" title="E125">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></i>;<br/>
<a NAME="E23:103_5"/><i><font color="Green" title="E126">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="#E14:103_5"><i><font color="Green" title="E120">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="E24:103_5"/><i><font color="Green" title="E127">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="#E13:103_5"><i><font color="Green" title="E119">A52</font></i></a>, <a class="txt" href="lexbfs.html#E18"><i><font color="Green" title="E17">Lm5</font></i></a></i>;<br/>
<a NAME="E25:103_5"/><i><font color="Green" title="E128">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="E26:103_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="#E14:103_5"><i><font color="Green" title="E120">A53</font></i></a>, <a class="txt" href="#E16:103_5"><i><font color="Green" title="E121">A54</font></i></a>, <a class="txt" href="#E17:103_5"><i><font color="Green" title="E122">A55</font></i></a>, <a class="txt" href="#E23:103_5"><i><font color="Green" title="E126">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="E27:103_5"/><b>then </b><i><font color="Green" title="E129">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="#E14:103_5"><i><font color="Green" title="E120">A53</font></i></a>, <a class="txt" href="#E25:103_5"><i><font color="Green" title="E128">A62</font></i></a></i>;<br/>
<a NAME="E28:103_5"/><i><font color="Green" title="E130">A64</font></i>: 
( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">b</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">a</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E23:103"><i><font color="Green" title="E94">A18</font></i></a>, <a class="txt" href="#E34:103"><i><font color="Green" title="E102">A34</font></i></a>, <a class="txt" href="#E38:103"><i><font color="Green" title="E104">A36</font></i></a>, <a class="txt" href="#E39:103"><i><font color="Green" title="E105">A37</font></i></a>, <a class="txt" href="#E41:103"><i><font color="Green" title="E106">A38</font></i></a>, <a class="txt" href="#E43:103"><i><font color="Green" title="E107">A39</font></i></a>, <a class="txt" href="#E44:103"><i><font color="Green" title="E108">A40</font></i></a>, <a class="txt" href="#E2:103_5"><i><font color="Green" title="E111">A44</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  implies <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font> )</span><br/><b>assume </b><a NAME="E1:103_5_2"/><i><font color="Green" title="E131">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font></span><br/><a NAME="E2:103_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="#E1:103_5_2"><i><font color="Green" title="E131">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="E3:103_5_2"/><b>then </b><i><font color="Green" title="E132">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="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="#E12:103_5"><i><font color="Green" title="E118">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="103_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_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="#E3:103_5_2"><i><font color="Green" title="E132">A66</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_2_1_1"><b>suppose </b></a><a NAME="E1:103_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font></span><br/><div class="add"><a NAME="E2:103_5_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E18:103_5"><i><font color="Green" title="E123">A56</font></i></a></i>;<br/><b>hence </b><a NAME="E3:103_5_2_1_1"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E2:103_5"><i><font color="Green" title="E111">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_2_1_2"><b>suppose </b></a><a NAME="E1:103_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">c</font></span><br/><div class="add"><b>hence </b><a NAME="E2:103_5_2_1_2"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E30:103_5"/>
<font color="Maroon">S<sub>2</sub></font>[4]
 <b>by </b><i><a class="txt" href="#E4:103_5"><i><font color="Green" title="E113">A46</font></i></a>, <a class="txt" href="#E22:103_5"><i><font color="Green" title="E125">A59</font></i></a>, <a class="txt" href="#E24:103_5"><i><font color="Green" title="E127">A61</font></i></a>, <a class="txt" href="#E27:103_5"><i><font color="Green" title="E129">A63</font></i></a>, <a class="txt" href="#E28:103_5"><i><font color="Green" title="E130">A64</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E47:103"/><i><font color="Green" title="E111">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="103_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">kk</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">kk</font>] implies <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] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:103_6"/><i><font color="Green" title="E112">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="E2:103_6"/><i><font color="Green" title="E113">A69</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">kk</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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]</span><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="#E1:103_6"><i><font color="Green" title="E112">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="E5:103_6"/><i><font color="Green" title="E114">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="E6:103_6"/><i><font color="Green" title="E115">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="E7:103_6"/><i><font color="Green" title="E116">A72</font></i>: 
( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">v3</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v2</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="E8:103_6"/><i><font color="Green" title="E117">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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="E9:103_6"/><i><font color="Green" title="E118">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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E2:103_6"><i><font color="Green" title="E113">A69</font></i></a></i>;<br/>
<a NAME="E10:103_6"/><i><font color="Green" title="E119">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="#E5:103_6"><i><font color="Green" title="E114">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="E11:103_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="#E1:103_6"><i><font color="Green" title="E112">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="E12:103_6"/><b>then </b><i><font color="Green" title="E120">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="E13:103_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="#E10:103_6"><i><font color="Green" title="E119">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="E15:103_6"/><i><font color="Green" title="E121">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="#E5:103_6"><i><font color="Green" title="E114">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="103_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <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></span><br/><b>assume </b><a NAME="E1:103_6_1"/><i><font color="Green" title="E122">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:103_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="#E10:103_6"><i><font color="Green" title="E119">A75</font></i></a>, <a class="txt" href="#E12:103_6"><i><font color="Green" title="E120">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="E3:103_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="#E5:103_6"><i><font color="Green" title="E114">A70</font></i></a>, <a class="txt" href="#E6:103_6"><i><font color="Green" title="E115">A71</font></i></a>, <a class="txt" href="#E1:103_6_1"><i><font color="Green" title="E122">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="E4:103_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:103_6"><i><font color="Green" title="E112">A68</font></i></a>, <a class="txt" href="#E10:103_6"><i><font color="Green" title="E119">A75</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E17:103_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="#E6:103_6"><i><font color="Green" title="E115">A71</font></i></a>, <a class="txt" href="#E15:103_6"><i><font color="Green" title="E121">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="E19:103_6"/><i><font color="Green" title="E122">A79</font></i>: 
( <font color="Maroon" title="c28">v5</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">L</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="E20:103_6"/><i><font color="Green" title="E123">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="E21:103_6"/><i><font color="Green" title="E124">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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="#E2:103"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="#E7:103_6"><i><font color="Green" title="E116">A72</font></i></a>, <a class="ref" href="lexbfs.html#D18" target="_self" title="LEXBFS:def.18">Def34</a></i>;<br/>
<div><i><font color="Green" title="E125">A82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_6_2"/><i><font color="Green" title="E126">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:103_6_2"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E7:103_6"><i><font color="Green" title="E116">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="E3:103_6_2"/><b>then </b>
( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v2</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E7:103_6"><i><font color="Green" title="E116">A72</font></i></a>, <a class="txt" href="#E19:103_6"><i><font color="Green" title="E122">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="E5:103_6_2"/><i><font color="Green" title="E127">A84</font></i>: 
( <font color="Maroon" title="c29">v6</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">L</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="E6:103_6_2"/><i><font color="Green" title="E128">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="E7:103_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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="#E2:103"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="#E20:103_6"><i><font color="Green" title="E123">A80</font></i></a>, <a class="txt" href="#E1:103_6_2"><i><font color="Green" title="E126">A83</font></i></a>, <a class="ref" href="lexbfs.html#D18" target="_self" title="LEXBFS:def.18">Def34</a></i>;<br/><b>thus </b><a NAME="E8:103_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:103_6"><i><font color="Green" title="E118">A74</font></i></a>, <a class="txt" href="#E19:103_6"><i><font color="Green" title="E122">A79</font></i></a>, <a class="txt" href="#E5:103_6_2"><i><font color="Green" title="E127">A84</font></i></a>, <a class="txt" href="#E6:103_6_2"><i><font color="Green" title="E128">A85</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="E24:103_6"/><i><font color="Green" title="E126">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="#E6:103_6"><i><font color="Green" title="E115">A71</font></i></a>, <a class="txt" href="#E20:103_6"><i><font color="Green" title="E123">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="E25:103_6"/><i><font color="Green" title="E127">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="#E5:103_6"><i><font color="Green" title="E114">A70</font></i></a>, <a class="txt" href="#E24:103_6"><i><font color="Green" title="E126">A86</font></i></a>, <a class="txt" href="lexbfs.html#E17"><i><font color="Green" title="E16">Lm4</font></i></a></i>;<br/>
<a NAME="E26:103_6"/><i><font color="Green" title="E128">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="#E24:103_6"><i><font color="Green" title="E126">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="E27:103_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="#E10:103_6"><i><font color="Green" title="E119">A75</font></i></a>, <a class="txt" href="#E12:103_6"><i><font color="Green" title="E120">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="E28:103_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="#E6:103_6"><i><font color="Green" title="E115">A71</font></i></a>, <a class="txt" href="#E26:103_6"><i><font color="Green" title="E128">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="E29:103_6"/><b>then </b><i><font color="Green" title="E129">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="#E24:103_6"><i><font color="Green" title="E126">A86</font></i></a>, <a class="txt" href="#E26:103_6"><i><font color="Green" title="E128">A88</font></i></a>, <a class="txt" href="lexbfs.html#E19"><i><font color="Green" title="E18">Lm6</font></i></a></i>;<br/>
<a NAME="E30:103_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="#E10:103_6"><i><font color="Green" title="E119">A75</font></i></a>, <a class="txt" href="#E12:103_6"><i><font color="Green" title="E120">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="E31:103_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="E32:103_6"/><b>then </b><i><font color="Green" title="E130">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="#E6:103_6"><i><font color="Green" title="E115">A71</font></i></a>, <a class="txt" href="#E24:103_6"><i><font color="Green" title="E126">A86</font></i></a>, <a class="txt" href="lexbfs.html#E19"><i><font color="Green" title="E18">Lm6</font></i></a></i>;<br/>
<a NAME="E33:103_6"/><i><font color="Green" title="E131">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="#E6:103_6"><i><font color="Green" title="E115">A71</font></i></a>, <a class="txt" href="#E24:103_6"><i><font color="Green" title="E126">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="E34:103_6"/><i><font color="Green" title="E132">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="#E8:103_6"><i><font color="Green" title="E117">A73</font></i></a>, <a class="txt" href="#E19:103_6"><i><font color="Green" title="E122">A79</font></i></a></i>;<br/>
<a NAME="E35:103_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="#E6:103_6"><i><font color="Green" title="E115">A71</font></i></a>, <a class="txt" href="#E22:103_6"><i><font color="Green" title="E125">A82</font></i></a>, <a class="ref" href="chord.html#D3" title="CHORD:def.3">CHORD:def 3</a></i>;<br/>
<a NAME="E36:103_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="#E5:103_6"><i><font color="Green" title="E114">A70</font></i></a>, <a class="txt" href="#E24:103_6"><i><font color="Green" title="E126">A86</font></i></a>, <a class="txt" href="#E34:103_6"><i><font color="Green" title="E132">A92</font></i></a>, <a class="ref" href="chord.html#T97" title="CHORD:th.97">CHORD:97</a></i>;<br/>
<a NAME="E37:103_6"/><b>then </b><i><font color="Green" title="E133">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></i>;<br/>
<a NAME="E38:103_6"/><i><font color="Green" title="E134">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="#E25:103_6"><i><font color="Green" title="E127">A87</font></i></a>, <a class="txt" href="lexbfs.html#E18"><i><font color="Green" title="E17">Lm5</font></i></a></i>;<br/>
<a NAME="E39:103_6"/><i><font color="Green" title="E135">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="E40:103_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="#E29:103_6"><i><font color="Green" title="E129">A89</font></i></a>, <a class="txt" href="#E32:103_6"><i><font color="Green" title="E130">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="E41:103_6"/><b>then </b><i><font color="Green" title="E136">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="#E33:103_6"><i><font color="Green" title="E131">A91</font></i></a>, <a class="txt" href="#E39:103_6"><i><font color="Green" title="E135">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="103_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  implies <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font> )</span><br/><b>assume </b><a NAME="E1:103_6_3"/><i><font color="Green" title="E137">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font></span><br/><a NAME="E2:103_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="#E1:103_6_3"><i><font color="Green" title="E137">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="E3:103_6_3"/><b>then </b><i><font color="Green" title="E138">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="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="#E24:103_6"><i><font color="Green" title="E126">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="103_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_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="#E3:103_6_3"><i><font color="Green" title="E138">A98</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_6_3_1_1"><b>suppose </b></a><a NAME="E1:103_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font></span><br/><div class="add"><a NAME="E2:103_6_3_1_1"/><b>then </b>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E8:103_6"><i><font color="Green" title="E117">A73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:103_6_3_1_1"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="#E19:103_6"><i><font color="Green" title="E122">A79</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_6_3_1_2"><b>suppose </b></a><a NAME="E1:103_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">v5</font></span><br/><div class="add"><b>hence </b><a NAME="E2:103_6_3_1_2"/>
<font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E43:103_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="#E7:103_6"><i><font color="Green" title="E116">A72</font></i></a>, <a class="txt" href="#E19:103_6"><i><font color="Green" title="E122">A79</font></i></a>, <a class="txt" href="#E21:103_6"><i><font color="Green" title="E124">A81</font></i></a>, <a class="txt" href="#E37:103_6"><i><font color="Green" title="E133">A93</font></i></a>, <a class="txt" href="#E38:103_6"><i><font color="Green" title="E134">A94</font></i></a>, <a class="txt" href="#E41:103_6"><i><font color="Green" title="E136">A96</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E48:103"/><i><font color="Green" title="E112">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="#E46:103"><i><font color="Green" title="E110">A43</font></i></a>, <a class="txt" href="#E47:103"><i><font color="Green" title="E111">A67</font></i></a>);<br/></i>
<a NAME="E49:103"/>
( 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="E50:103"/><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="#E48:103"><i><font color="Green" title="E112">A99</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="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="E52:103"/><i><font color="Green" title="E113">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="E53:103"/><i><font color="Green" title="E114">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="E54:103"/>
( <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="E55:103"/>
( <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">v3</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">v2</font> &amp; <font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <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="E56:103"/>
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/><font color="Maroon" title="c2">L</font> <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> <font color="Maroon" title="c2">L</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">v4</font>
 ;<br/>
<a NAME="E57:103"/>
 <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="#E53:103"><i><font color="Green" title="E114">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="E58:103"/><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="E59:103"/><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="#E52:103"><i><font color="Green" title="E113">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="E60:103"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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