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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-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">C</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 iff  not <font color="Maroon" title="c1">C</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  )</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"> (  not <font color="Maroon" title="c1">C</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c1">C</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:97_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.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">  not <font color="Maroon" title="c1">C</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span><br/><a NAME="E2:97_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A2</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 <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>;<br/><span class="kw">consider </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:97_1"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> &amp;  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="card_2.html#T60" onmouseover="rs('card_2/T60')" onmouseout="rh()">CARD_2:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c5">v2</font> = <font color="Maroon" title="c3">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>2 = 
<font color="Maroon" title="c4">v1</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_016.html#D4" onmouseover="rs('glib_016/D4')" onmouseout="rh()">GLIB_016:def 4</a></span>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</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="#E2:97_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="glib_000.html#T19" onmouseover="rs('glib_000/T19')" onmouseout="rh()">GLIB_000:19</a></span>
;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">e1</font>, <font color="Maroon" title="c7">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:97_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
( <font color="Maroon" title="c6">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e2</font> &amp; <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c7">e2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T60" onmouseover="rs('card_2/T60')" onmouseout="rh()">CARD_2:60</a></span>;<br/><a NAME="E9:97_1"/>
<font color="Maroon" title="c6">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</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="#E8:97_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E11:97_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c6">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/><a NAME="E12:97_1"/>
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</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="#E8:97_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E14:97_1"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c9">w2</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/><a NAME="E15:97_1"/>
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">w1</font> &amp; <font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E11:97_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E14:97_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</a></span>;<br/><a NAME="E16:97_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> &amp; <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <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="E17:97_1"/>
 not <font color="Maroon" title="c1">C</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E11:97_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E14:97_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="glib_000.html#D20" onmouseover="rs('glib_000/D20')" onmouseout="rh()">GLIB_000:def 20</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="E2:97"/>
 not <font color="Maroon" title="c1">C</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</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">C</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">e1</font>, <font color="Maroon" title="c3">e2</font>, <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:97"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
( <font color="Maroon" title="c2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c2">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">e2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D20" onmouseover="rs('glib_000/D20')" onmouseout="rh()">GLIB_000:def 20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c7">v2</font> = <font color="Maroon" title="c5">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">e1</font> = <font color="Maroon" title="c2">e1</font>, <font color="Maroon" title="c9">e2</font> = <font color="Maroon" title="c3">e2</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">W1</font> = <font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">W2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font>;<br/>
<a NAME="E7:97"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</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="c6">v1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</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="c7">v2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/>
<a NAME="E8:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/>
<a NAME="E9:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a>, <a class="ref" href="glib_001.html#T119" onmouseover="rs('glib_001/T119')" onmouseout="rh()">GLIB_001:119</a></span>;<br/>
<a NAME="E10:97"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_001.html#T14" onmouseover="rs('glib_001/T14')" onmouseout="rh()">GLIB_001:14</a></span>;<br/>
<a NAME="E11:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="glib_001.html#T64" onmouseover="rs('glib_001/T64')" onmouseout="rh()">GLIB_001:64</a></span>;<br/>
<a NAME="E12:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" onmouseover="rs('glib_001/T126')" onmouseout="rh()">GLIB_001:126</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c9">e2</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="glib_001.html#T82" onmouseover="rs('glib_001/T82')" onmouseout="rh()">GLIB_001:82</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c9">e2</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_001.html#T83" onmouseover="rs('glib_001/T83')" onmouseout="rh()">GLIB_001:83</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D9" onmouseover="rs('finseq_1/D9')" onmouseout="rh()">FINSEQ_1:def 9</a></span>
;<br/>
<a NAME="E14:97"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="finseq_3.html#T94" onmouseover="rs('finseq_3/T94')" onmouseout="rh()">FINSEQ_3:94</a></span>;<br/>
<a NAME="E15:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D27" onmouseover="rs('glib_001/D27')" onmouseout="rh()">GLIB_001:def 27</a></span>;<br/>
<a NAME="E16:97"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:97_3"/>
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</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="E2:97_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A15</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E3:97_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp; <font color="Maroon" title="c9">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T57" onmouseover="rs('glib_000/T57')" onmouseout="rh()">GLIB_000:57</a></span>;<br/>
<a NAME="E4:97_3"/><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"><font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">v1</font> <a href="glib_013.html#K3" title="GLIB_013:func.3">.inDegree()</a> 
 
<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="E5:97_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A16</font></span>: 
2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">v1</font> <a href="glib_013.html#K3" title="GLIB_013:func.3">.inDegree()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</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="E6:97_3"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="ref" href="glib_000.html#T59" onmouseover="rs('glib_000/T59')" onmouseout="rh()">GLIB_000:59</a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<a NAME="E7:97_3"/><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#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">v1</font> <a href="glib_013.html#K4" title="GLIB_013:func.4">.outDegree()</a> 
 
<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></span>;<br/>
<a NAME="E8:97_3"/><span class="kw">then </span>
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">v1</font> <a href="glib_013.html#K4" title="GLIB_013:func.4">.outDegree()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>;<br/>
<a NAME="E9:97_3"/><span class="kw">then </span>
2 <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">v1</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="#E5:97_3"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="card_2.html#T83" onmouseover="rs('card_2/T83')" onmouseout="rh()">CARD_2:83</a></span>;<br/>
<a NAME="E10:97_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A17</font></span>: 
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_016.html#D4" onmouseover="rs('glib_016/D4')" onmouseout="rh()">GLIB_016:def 4</a></span>;<br/>
<a NAME="E11:97_3"/>
2 <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="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E12:97_3"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:97_3"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:97_3"/>
contradiction
 ; <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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_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">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">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#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> holds <br/>(  not <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  not <font color="Olive" title="b2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c13">n</font> be   <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#NK1" title="NUMBERS:NK.1">NAT</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="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">n</font> &amp; <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> implies (  not <font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  not <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> ) )</span><br/><span class="lab"><font color="Green" title="E57">A18</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c9">e2</font>,<font color="Maroon" title="c6">v1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="glib_001.html#T62" onmouseover="rs('glib_001/T62')" onmouseout="rh()">GLIB_001:62</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c9">e2</font>,<font color="Maroon" title="c6">v1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_001.html#D5" onmouseover="rs('glib_001/D5')" onmouseout="rh()">GLIB_001:def 5</a></span>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K8" title="FINSEQ_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c9">e2</font>,<font color="Maroon" title="c6">v1</font></span><a href="finseq_4.html#K8" title="FINSEQ_4:func.8">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T75" onmouseover="rs('finseq_4/T75')" onmouseout="rh()">FINSEQ_4:75</a></span>
;<br/><span class="kw">assume </span><a NAME="E2:97_4"/><span class="lab"><font color="Green" title="E58">A19</font></span>: 
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">n</font> &amp; <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</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"> (  not <font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  not <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E3:97_4"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</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"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_4_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="E1:97_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_4_2_1"><span class="kw">suppose </span></a><a NAME="E1:97_4_2_1"/><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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"><a NAME="E2:97_4_2_1"/>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:97"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</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="E3:97_4_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A21</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_4_2_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="ref" href="chord.html#T7" onmouseover="rs('chord/T7')" onmouseout="rh()">CHORD:7</a></span>;<br/><a NAME="E4:97_4_2_1"/>
<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:97"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E5:97_4_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T8" onmouseover="rs('chord/T8')" onmouseout="rh()">CHORD:8</a></span>;<br/><a NAME="E6:97_4_2_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <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="c6">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_4"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="txt" href="#E3:97_4_2_1"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="ref" href="finseq_4.html#T78" onmouseover="rs('finseq_4/T78')" onmouseout="rh()">FINSEQ_4:78</a></span>;<br/><span class="kw">hence </span><a NAME="E7:97_4_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:97"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</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="97_4_2_2"><span class="kw">suppose </span></a><a NAME="E1:97_4_2_2"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</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"> contradiction</span><br/><div class="add"><a NAME="E2:97_4_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:97"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:97_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A22</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:97_4_2_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" onmouseover="rs('chord/T2')" onmouseout="rh()">CHORD:2</a></span>;<br/><a NAME="E5:97_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A23</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="txt" href="#E3:97_4_2_2"><span class="lab"><font color="Green" title="E59">A22</font></span></a>, <a class="ref" href="chord.html#T7" onmouseover="rs('chord/T7')" onmouseout="rh()">CHORD:7</a></span>;<br/><a NAME="E6:97_4_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</font></span></a></span>;<br/><a NAME="E7:97_4_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T6" onmouseover="rs('chord/T6')" onmouseout="rh()">CHORD:6</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:97_4_2_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <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="c7">v2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_4"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="txt" href="#E5:97_4_2_2"><span class="lab"><font color="Green" title="E60">A23</font></span></a>, <a class="ref" href="finseq_4.html#T78" onmouseover="rs('finseq_4/T78')" onmouseout="rh()">FINSEQ_4:78</a></span>;<br/><span class="kw">hence </span><a NAME="E9:97_4_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:97"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E2:97_4"><span class="lab"><font color="Green" title="E58">A19</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>
<a NAME="E18:97"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:97"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a></span>;<br/>
<span class="kw">then </span> <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">C</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e2</font></span>)</span> <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="#E9:97"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="txt" href="#E12:97"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="glpacy00.html#T31" target="_self" onmouseover="rs('glpacy00/T31')" onmouseout="rh()">Th39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="glib_001.html#T95" onmouseover="rs('glib_001/T95')" onmouseout="rh()">GLIB_001:95</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="glib_001.html#T91" onmouseover="rs('glib_001/T91')" onmouseout="rh()">GLIB_001:91</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T2" onmouseover="rs('enumset1/T2')" onmouseout="rh()">ENUMSET1:2</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></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="enumset1.html#T30" onmouseover="rs('enumset1/T30')" onmouseout="rh()">ENUMSET1:30</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E20:97"/>
<font color="Maroon" title="c1">C</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:97"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="card_2.html#T57" onmouseover="rs('card_2/T57')" onmouseout="rh()">CARD_2:57</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>
