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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>set </b><font color="Maroon">W</font> = <font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font>;<br/>
<a NAME="E1:38"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<b>hence </b><a NAME="E2:38"/>
<font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span>
 ;<br/><a NAME="E2:38_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><b>hence </b><a NAME="E3:38_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38"/>
<font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font> is <a href="glib_001.html#V2" target="_self">directed</a>
 <b>by </b><i/>;<br/>

<a NAME="E5:38"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:38"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E7:38"/><b>then </b>
0 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:38"/><b>then </b>
0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 
;<br/>
<a NAME="E9:38"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K15" target="_self">.length()</a>  <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E10:38"/>
<font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font> is <a href="glib_001.html#V3" target="_self">trivial</a>
 <b>by </b><i/>;<br/>

<a NAME="E11:38"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E12:38"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E13:38"/><b>then </b>
0 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E14:38"/><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">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E15:38"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E16:38"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T1" target="_self">Th1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:38_2"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> )
 ;<br/><a NAME="E2:38_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E3:38_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:38_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> implies ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:38"/>
<font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i>, </i>;<br/>


</div>
