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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> non <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c2">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>;<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> ex <font color="Olive" title="b1">W</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<div><i><font color="Green" title="E36">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c4">P</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green" title="E37">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 ;<br/><a NAME="E4:52_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a> </span>)</span> <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="ref" href="glib_001.html#T155" title="GLIB_001:th.155">GLIB_001:155</a></i>;<br/><a NAME="E5:52_1"/><b>then </b>
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="c4">P</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E6:52_1"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:52_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="glib_001.html#D14" title="GLIB_001:def.14">GLIB_001:def 14</a></i>;<br/><a NAME="E7:52_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E8:52_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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> 1</span>)</span>
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c3">src</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">tar</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:52"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:52"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/>
<a NAME="E4:52"/><b>then </b><i><font color="Green" title="E37">A4</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="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></i>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green" title="E38">A5</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>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c5">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:52"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] &amp; ( 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="c5">k</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="glib_002.html#E2:52"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="glib_002.html#E5:52"><i><font color="Green" title="E38">A5</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c6">W</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:52"/><i><font color="Green" title="E40">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> &amp; ( 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="c5">k</font> ) )
 <b>by </b><i><a class="txt" href="glib_002.html#E7:52"><i><font color="Green" title="E39">A6</font></i></a></i>;<br/>
<a NAME="E10:52"/><i><font color="Green" title="E41">A8</font></i>: 
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="c6">W</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E4:52"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="glib_002.html#E9:52"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/>
<a NAME="E11:52"/><b>then </b><i><font color="Green" title="E42">A9</font></i>: 
not <font color="Maroon" title="c6">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/>
<div><i><font color="Green" title="E43">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon" title="c6">W</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="c6">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><a NAME="E2:52_2"/><b>then </b>
<font color="Maroon" title="c6">W</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><a NAME="E3:52_2"/><b>then </b>
<font color="Maroon" title="c6">W</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E11:52"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E4:52_2"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self" title="GLIB_002:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:52"/><b>then </b><i><font color="Green" title="E44">A11</font></i>: 
not <font color="Maroon" title="c6">W</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/>
<div><i><font color="Green" title="E45">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">W</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:52_3"/><i><font color="Green" title="E46">A13</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> &amp; not <font color="Maroon" title="c7">W</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> )
 ;<br/><a NAME="E2:52_3"/><b>then </b><i><font color="Green" title="E47">A14</font></i>: 
not <font color="Maroon" title="c7">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E9:52"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="glib_002.html#E10:52"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/><a NAME="E3:52_3"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:52"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="glib_002.html#E9:52"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="glib_002.html#E1:52_3"><i><font color="Green" title="E46">A13</font></i></a></i>;<br/><a NAME="E4:52_3"/><b>then </b>
2 <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="c7">W</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">lenW2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E6:52_3"/><i><font color="Green" title="E48">A15</font></i>: 
<font color="Maroon" title="c8">lenW2</font> <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="c7">W</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_3_1"/>
not <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 ;<br/><a NAME="E2:52_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="glib_000.html#D54" title="GLIB_000:def.54">GLIB_000:def 54</a></i>;<br/><a NAME="E3:52_3_1"/><b>then </b><i><font color="Green" title="E49">A16</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="glib_000.html#T22" title="GLIB_000:th.22">GLIB_000:22</a></i>;<br/><a NAME="E4:52_3_1"/>
<font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><a NAME="E5:52_3_1"/><b>then </b>
not <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:52_3"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="glib_001.html#T136" title="GLIB_001:th.136">GLIB_001:136</a></i>;<br/><a NAME="E6:52_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D52" title="GLIB_000:def.52">GLIB_000:def 52</a></i>;<br/><a NAME="E7:52_3_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T22" title="GLIB_000:th.22">GLIB_000:22</a></i>;<br/><a NAME="E8:52_3_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E9:52_3_1"/><b>then </b>
<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>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <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="E10:52_3_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:52_3_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">e1</font>, <font color="Maroon" title="c10">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:52_3_1"/><i><font color="Green" title="E50">A17</font></i>: 
( <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  &amp; <font color="Maroon" title="c10">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  &amp; <font color="Maroon" title="c9">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">e2</font> )
 <b>by </b><i><a class="ref" href="glib_002.html#T1" target="_self" title="GLIB_002:th.1">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_1_1"/>
( <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c9">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_1_1_1"/><i><font color="Green" title="E51">A18</font></i>: 
<font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">e2</font> = <font color="Maroon" title="c10">e2</font>;<br/><b>thus </b><a NAME="E2:52_3_1_1_1_1"/>
( <font color="Maroon" title="c11">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  &amp; <font color="Maroon" title="c11">e2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E12:52_3_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="glib_002.html#E1:52_3_1_1_1_1"><i><font color="Green" title="E51">A18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_1_1_2"/><i><font color="Green" title="E51">A19</font></i>: 
<font color="Maroon" title="c9">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">e1</font> = <font color="Maroon" title="c9">e1</font>;<br/><b>thus </b><a NAME="E2:52_3_1_1_1_2"/>
( <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  &amp; <font color="Maroon" title="c11">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E12:52_3_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="glib_002.html#E1:52_3_1_1_1_2"><i><font color="Green" title="E51">A19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c11">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:52_3_1"/><i><font color="Green" title="E51">A20</font></i>: 
( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  &amp; <font color="Maroon" title="c11">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ;<br/><b>consider </b><font color="Maroon" title="c12">v</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="E17:52_3_1"/><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E15:52_3_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="glib_000.html#T67" title="GLIB_000:th.67">GLIB_000:67</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_2_1"/>
( <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or  not <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1"/>
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:52_3_1_2_1_1"/><i><font color="Green" title="E53">A22</font></i>: 
( <font color="Maroon" title="c13">n</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="c7">W</font> &amp; <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><b>set </b><font color="Maroon" title="c14">m</font> = <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font>;<br/><a NAME="E4:52_3_1_2_1_1"/><i><font color="Green" title="E54">A23</font></i>: 
( <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</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="c7">W</font> &amp; <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <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="c7">W</font> &amp; <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font> ) )
 <b>by </b><i><a class="txt" href="glib_002.html#E3:52_3_1_2_1_1"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="glib_001.html#D22" title="GLIB_001:def.22">GLIB_001:def 22</a></i>;<br/><b>set </b><font color="Maroon" title="c15">W2</font> = <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span>;<br/><a NAME="E5:52_3_1_2_1_1"/><i><font color="Green" title="E55">A24</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <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="c12">v</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></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="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E4:52_3_1_2_1_1"><i><font color="Green" title="E54">A23</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E6:52_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E56">A25</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E17:52_3_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>;<br/><a NAME="E7:52_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E57">A26</font></i>: 
<font color="Maroon" title="c11">e</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="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T65" title="GLIB_000:th.65">GLIB_000:65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_2_1_1_1_1"/>
(  not <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  or <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_1"/>
not <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E58">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:52_3_1_2_1_1"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="glib_002.html#E34"><i><font color="Green" title="E33">Lm22</font></i></a></i>;<br/><a NAME="E3:52_3_1_2_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font></span>)</span> <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="c12">v</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">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="c12">v</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E5:52_3_1_2_1_1"><i><font color="Green" title="E55">A24</font></i></a>, <a class="txt" href="glib_002.html#E6:52_3_1_2_1_1"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></i>;<br/><a NAME="E4:52_3_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E59">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><a NAME="E5:52_3_1_2_1_1_1_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:52_3_1_2_1_1"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="glib_001.html#T133" title="GLIB_001:th.133">GLIB_001:133</a></i>;<br/><a NAME="E6:52_3_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:52_3_1_2_1_1_1_1_1"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="glib_002.html#E4:52_3_1_2_1_1_1_1_1"><i><font color="Green" title="E59">A28</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E7:52_3_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self" title="GLIB_002:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_2"/><i><font color="Green" title="E58">A29</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T107" title="GLIB_001:th.107">GLIB_001:107</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">a</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:52_3_1_2_1_1_1_1_2"/><i><font color="Green" title="E59">A30</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font> &amp; <font color="Maroon" title="c16">a</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="c7">W</font> &amp; <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E58">A29</font></i></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><a NAME="E5:52_3_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="glib_002.html#E4:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E6:52_3_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">a</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:52_3_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:52_3_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E60">A31</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E15:52_3_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="glib_002.html#E4:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">a1</font> = <font color="Maroon" title="c16">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_002.html#E4:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E10:52_3_1_2_1_1_1_1_2"/><i><font color="Green" title="E61">A32</font></i>: 
<font color="Maroon" title="c17">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E4:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E11:52_3_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font>
 ;<br/><a NAME="E12:52_3_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E62">A33</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a1</font>,<font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E59">A30</font></i></a>, <a class="txt" href="glib_002.html#E10:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E61">A32</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_1_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c7">W</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="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a1</font> &amp; <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> ) or ( <font color="Maroon" title="c7">W</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="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a1</font> ) )
 </span><b>by </b><i><a class="txt" href="glib_002.html#E17:52_3_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="glib_002.html#E12:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E62">A33</font></i></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">W</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="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a1</font> &amp; <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>;<br/><b>hence </b><a NAME="E3:52_3_1_2_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:52_3"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="glib_002.html#E10:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E61">A32</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c7">W</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="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a1</font> )
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E63">A34</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>;<br/><a NAME="E3:52_3_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c17">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">lenW2</font> <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="glib_002.html#E8:52_3_1_2_1_1_1_1_2"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E4:52_3_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c7">W</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:52_3"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:52_3_1_2_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:52_3"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="glib_002.html#E2:52_3_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E63">A34</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:52_3_1_2_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:52_3_1_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:52_3_1_2_1_2"/>
not <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:52_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E53">A35</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:52_3"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="glib_002.html#E17:52_3_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="glib_001.html#T152" title="GLIB_001:th.152">GLIB_001:152</a></i>;<br/><a NAME="E3:52_3_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="glib_002.html#E1:52_3"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="glib_002.html#E17:52_3_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/><a NAME="E4:52_3_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E9:52"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="glib_002.html#E2:52_3_1_2_1_2"><i><font color="Green" title="E53">A35</font></i></a></i>;<br/><b>hence </b><a NAME="E5:52_3_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:52_3_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:52_3"/>
<font color="Maroon" title="c7">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E15:52"/><b>then </b><i><font color="Green" title="E46">A36</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E9:52"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="glib_002.html#E13:52"><i><font color="Green" title="E44">A11</font></i></a></i>;<br/>
<a NAME="E16:52"/><i><font color="Green" title="E47">A37</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="c6">W</font> <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> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E9:52"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="glib_001.html#T22" title="GLIB_001:th.22">GLIB_001:22</a></i>;<br/>
<a NAME="E17:52"/>
not <font color="Maroon" title="c6">W</font> <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>
 
<b>by </b><i><a class="txt" href="glib_002.html#E13:52"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="glib_001.html#T121" title="GLIB_001:th.121">GLIB_001:121</a></i>;<br/>
<a NAME="E18:52"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W</font> <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>  is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E14:52"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="glib_002.html#E16:52"><i><font color="Green" title="E47">A37</font></i></a></i>;<br/>
<a NAME="E19:52"/><b>then </b><i><font color="Green" title="E48">A38</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></i>;<br/>
<a NAME="E20:52"/>
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c6">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E21:52"/><b>then </b>
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E22:52"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</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">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b1">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> &amp; <font color="Olive" title="b2">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b1">v1</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E12:52"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="glib_002.html#E15:52"><i><font color="Green" title="E46">A36</font></i></a>, <a class="txt" href="glib_002.html#E19:52"><i><font color="Green" title="E48">A38</font></i></a></i>;<br/>


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