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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_002.html#V2" target="_self">acyclic</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#NM5">Path</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>assume </b><a NAME="E1:51"/><i><font color="Green">E35</font></i>: 
( not <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>  &amp; <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>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a>  )
 ;<br/>

<a NAME="E2:51"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T70">GLIB_000:70</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:51"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V4">Trail-like</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:51"><i><font color="Green">E35</font></i></a>, <a class="ref" href="glib_001.html#T143">GLIB_001:143</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="glib_002.html#E2:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V3">trivial</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V3">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V3">trivial</a>
 ;<br/><div class="add"><a NAME="E2:51_1_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="glib_001.html#T127">GLIB_001:127</a></i>;<br/><b>hence </b><a NAME="E3:51_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:51"><i><font color="Green">E35</font></i></a>, <a class="txt" href="glib_002.html#E2:51"><i><font color="Green">E36</font></i></a>, <a class="txt" href="glib_002.html#E1:51_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="glib_001.html#T151">GLIB_001:151</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_2"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V3">trivial</a>
 ;<br/><div class="add"><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V1">closed</a>
 ;<br/><a NAME="E2:51_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:51_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E3:51_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:51_1_1_2_2"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_1_2_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<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>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><a NAME="E2:51_1_1_2_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>6</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>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:51"><i><font color="Green">E38</font></i></a>, <a class="txt" href="glib_002.html#E1:51_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:51_1_1_2_2_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<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>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="glib_001.html#V4">Trail-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_001.html#T142">GLIB_001:142</a></i>;<br/><a NAME="E4:51_1_1_2_2_1"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</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>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E2:51_1_1_2_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E5:51_1_1_2_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:51_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:51_1_1_2_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:51_1_1_2_2_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_002.html#E4:51_1_1_2_2_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="glib_002.html#E2:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K4">.last()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
<b>by </b><i><a class="txt" href="glib_002.html#E4:51_1_1_2_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="glib_002.html#E2:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<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>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="glib_001.html#V3">trivial</a>
 ;<br/><a NAME="E2:51_1_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#T127">GLIB_001:127</a></i>;<br/><a NAME="E3:51_1_1_2_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E2:51_1_1_2_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E4:51_1_1_2_2_1_2"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:51"><i><font color="Green">E38</font></i></a>, <a class="txt" href="glib_002.html#E1:51_1_1_2_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E5:51_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:51_1_1_2_2_1_2"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E7:51_1_1_2_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>6</sub></font>,<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>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:51_1_1_2_2_1"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:51_1_1_2_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="glib_001.html#T167">GLIB_001:167</a></i>;<br/><a NAME="E11:51_1_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:51_1_1_2_2_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#E7:51_1_1_2_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_001.html#D32">GLIB_001:def 32</a></i>;<br/><a NAME="E12:51_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T120">GLIB_001:120</a></i>;<br/><a NAME="E13:51_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E10:51_1_1_2_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E14:51_1_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:51_1_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:51_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:51"><i><font color="Green">E35</font></i></a>, <a class="txt" href="glib_002.html#E2:51"><i><font color="Green">E36</font></i></a>, <a class="txt" href="glib_002.html#E2:51_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="glib_001.html#T151">GLIB_001:151</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:51"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V5">Path-like</a>
 ;<br/>


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