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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" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:85_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V2" target="_self">directed</a>
 ;<br/><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="E2:85_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E3:85_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E4:85_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">W</font> <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"><font color="Maroon">W</font> <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>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:85_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:85_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a>, </i>;<br/><b>hence </b><a NAME="E3:85_1_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:85_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:85"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_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> ;<br/><b>assume </b><a NAME="E1:85_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E2:85_2"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:85_2"/>
<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"><font color="Maroon">W</font> <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> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:85"/>
<font color="Maroon">W</font> is <a href="glib_001.html#V2" target="_self">directed</a>
 <b>by </b><i/>;<br/>


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