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

<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 implies <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 implies <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:59_1"/>
<font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c1">P</font> is 2 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Olive" title="b2">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  ) )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c2">v</font> =  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>;<br/><span class="kw">take </span><font color="Maroon" title="c3">v</font> =  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b1">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Olive" title="b1">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E1:59_1_1"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Olive" title="b1">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">w</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> </span><br/><span class="kw">thus </span><a NAME="E2:59_1_1"/>
<font color="Maroon" title="c4">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:59_1"/>
<font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#T48" onmouseover="rs('glib_013/T48')" onmouseout="rh()">GLIB_013:48</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:59"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_2"/>
(  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>  holds  not <font color="Olive" title="b1">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or  ex <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> st <font color="Olive" title="b1">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1"><span class="kw">suppose </span></a><a NAME="E1:59_2_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>  holds  not <font color="Olive" title="b1">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; (  for <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Olive" title="b2">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  ) )</span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c2">v</font> =  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; (  for <font color="Olive" title="b1">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Olive" title="b1">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  ) )</span><br/><a NAME="E1:59_2_1_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_2_1_1_1"/>
 not 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:59_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a>, <a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a></span>;<br/>
<a NAME="E3:59_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:59_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="glib_000.html#T157" onmouseover="rs('glib_000/T157')" onmouseout="rh()">GLIB_000:157</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:59_2_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:59_2_1_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Olive" title="b1">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">w</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> </span><br/><span class="kw">thus </span><a NAME="E4:59_2_1_1"/>
<font color="Maroon" title="c3">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">Def1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:59_2_1"/>
<font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#T48" onmouseover="rs('glib_013/T48')" onmouseout="rh()">GLIB_013:48</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_2"><span class="kw">suppose </span></a><a NAME="E1:59_2_2"/>
 ex <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> st <font color="Olive" title="b1">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E3:59_2_2"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c2">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ;<br/><a NAME="E4:59_2_2"/>
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_2_2_1"/>
 not 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:59_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a></span>;<br/>
<a NAME="E3:59_2_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">P</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:59_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E6:59_2_2"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
( <font color="Maroon" title="c3">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c4">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c2">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="glibpre0.html#T95" onmouseover="rs('glibpre0/T95')" onmouseout="rh()">GLIBPRE0:95</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:59_2_2"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_2_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E10:59_2_2"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<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="c3">u</font>,<font color="Maroon" title="c4">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">T</font> =  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font>;<br/><a NAME="E11:59_2_2"/>
 the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <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="c3">u</font>,<font color="Maroon" title="c4">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59_2_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="glib_001.html#T160" onmouseover="rs('glib_001/T160')" onmouseout="rh()">GLIB_001:160</a></span>;<br/><a NAME="E12:59_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
(  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <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="c3">u</font> &amp;  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">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="c4">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/><a NAME="E13:59_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A12</font></span>: 
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>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_2_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="glib_001.html#T127" onmouseover="rs('glib_001/T127')" onmouseout="rh()">GLIB_001:127</a>, <a class="ref" href="glib_001.html#T125" onmouseover="rs('glib_001/T125')" onmouseout="rh()">GLIB_001:125</a></span>;<br/><a NAME="E14:59_2_2"/>
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>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_2_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E15:59_2_2"/><span class="kw">then </span>
 the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T4" onmouseover="rs('polyform/T4')" onmouseout="rh()">POLYFORM:4</a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E16:59_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A13</font></span>: 
 the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">u</font>, the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59_2_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>;<br/><a NAME="E17:59_2_2"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
 the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_2_2_2"/>
 the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</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="c2">v</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:59_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_2_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a>, <a class="ref" href="polyform.html#T6" onmouseover="rs('polyform/T6')" onmouseout="rh()">POLYFORM:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:59_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E6:59_2_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E12:59_2_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="glib_001.html#T143" onmouseover="rs('glib_001/T143')" onmouseout="rh()">GLIB_001:143</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:59_2_2"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_2_2_3"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_2_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_2_2_3_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 ) or ( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_2_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E16:59_2_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_2_2_3_1_1"/>
( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:59_2_2_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59_2_2"><span class="lab"><font color="Green" title="E44">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_2_2_3_1_2"/>
( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:59_2_2_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59_2_2"><span class="lab"><font color="Green" title="E44">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2_2_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Olive" title="b2">x</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  ) )</span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c3">u</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; (  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">P</font> holds  <font color="Olive" title="b1">x</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  ) )</span><br/><a NAME="E1:59_2_2_4"/>
(  the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">u</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  &amp; <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">u</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_2_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E16:59_2_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</a></span>;<br/><a NAME="E2:59_2_2_4"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"> the   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c5">e</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_1:11</a>, <a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a></span>;<br/><a NAME="E3:59_2_2_4"/><span class="kw">then </span>
2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_2_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="card_2.html#T57" onmouseover="rs('card_2/T57')" onmouseout="rh()">CARD_2:57</a></span>;<br/><a NAME="E4:59_2_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A17</font></span>: 
2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T19" onmouseover="rs('glib_000/T19')" onmouseout="rh()">GLIB_000:19</a></span>;<br/><a NAME="E5:59_2_2_4"/>
<font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:59_2_2_4"/>
<font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2_2_4"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">P</font> holds  <font color="Olive" title="b1">x</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> </span><br/><a NAME="E7:59_2_2_4"/>
<font color="Maroon" title="c9">x</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:59_2_2_4"/>
<font color="Maroon" title="c9">x</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2_2_4"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E20:59_2_2"/>
<font color="Maroon" title="c1">P</font> <a href="glib_013.html#K18" title="GLIB_013:func.18">.maxDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#T48" onmouseover="rs('glib_013/T48')" onmouseout="rh()">GLIB_013:48</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
