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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">W</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<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>assume </b><a NAME="E1:21"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:21"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1"/>
 ex <font color="Olive">W2</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font> st <font color="Olive">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E3:21_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_2"/>
 ex <font color="Olive">W2</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font> st <font color="Olive">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/>


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