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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_002.html#V1" target="_self">connected</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="glib_000.html#NM8">removeEdge</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:64"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V8">Cycle-like</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K14">.edges()</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E3:64"/>
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:64"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:64"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#K14">.edges()</a>  )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:64"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_001.html#T158">GLIB_001:158</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E5:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="glib_001.html#T173">GLIB_001:173</a></i>;<br/>
<a NAME="E7:64"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E5:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E1:64_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:64_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">c<sub>13</sub></font>;<br/><a NAME="E6:64_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:64_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E7:64_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K14">.edges()</a>  or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>17</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_1_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T104">GLIB_001:104</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="glib_000.html#T54">GLIB_000:54</a></i>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">c<sub>17</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>;<br/><a NAME="E5:64_1_1_1_1"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; not 1 is <a href="abian.html#V1">even</a> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/><a NAME="E6:64_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:64_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">c<sub>17</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:64_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E3:64_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#E5:64_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E8:64_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:64_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E3:64_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E9:64_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:64_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_1_1_1_1_1"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><a NAME="E4:64_1_1_1_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_002.html#E6:64_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_001.html#T46">GLIB_001:46</a></i>;<br/><a NAME="E5:64_1_1_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:64_1_1_1_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:64_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_002.html#E6:64_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><b>hence </b><a NAME="E8:64_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#E9:64_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="glib_002.html#E3:64_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_002.html#E5:64_1_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">c<sub>17</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T173">GLIB_001:173</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_1_1_1_1_2"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E5:64_1_1_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><a NAME="E6:64_1_1_1_1_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:64_1_1_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#E3:64_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_002.html#E5:64_1_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E8:64_1_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:64_1_1_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:64_1_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E11:64_1_1_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E9:64_1_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E12:64_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E13:64_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E3:64_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#E7:64_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="glib_002.html#E8:64_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="glib_002.html#E11:64_1_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T173">GLIB_001:173</a></i>;<br/><a NAME="E14:64_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:64_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E15:64_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E8:64_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_1_1_3_1"/>
( ( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) )
 </span><b>by </b><i><a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a>, <a class="txt" href="glib_002.html#E3:64_1_1_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:64_1_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>8</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:64"><i><font color="Green">E45</font></i></a>, <a class="txt" href="glib_002.html#E14:64_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><a NAME="E3:64_1_1_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>23</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E15:64_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1_1_1_3_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_3_1_2"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:64_1_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E8:64"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#E14:64_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><a NAME="E3:64_1_1_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#K7">.append</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>23</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E15:64_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1_1_1_3_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:64_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_1_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T173">GLIB_001:173</a></i>;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><b>thus </b><a NAME="E3:64_1_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:64_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:64_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:64"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/>


</div>
