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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>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">P</font></span>)</span> where <font color="Olive">P</font> is   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> : <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">T</font> being  <a href="glib_001.html#V5">Path-like</a>  <a href="glib_001.html#M4">Subwalk</a> of <font color="Maroon">W</font>;<br/>
<a NAME="E2:44"/>
( <font color="Maroon">T</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">T</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  )
 
<b>by </b><i><a class="ref" href="glib_001.html#T162">GLIB_001:162</a></i>;<br/>
<a NAME="E3:44"/><b>then </b>
<font color="Maroon">T</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E4:44"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">P</font></span>)</span> where <font color="Olive">P</font> is   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> : <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  </span>}</span> 
 
;<br/>
<a NAME="E5:44"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">P</font></span>)</span> where <font color="Olive">P</font> is   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> : <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<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="E1:44_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">P</font></span>)</span> where <font color="Olive">P</font> is   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> : <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  </span>}</span> 
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:44_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">P</font></span>)</span> where <font color="Olive">P</font> is   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> : <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:44"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:44"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E10:44"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 ;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E11:44"/>
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">W2</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D2" target="_self">CHORD:def 2</a><br/>

<b>assume </b><a NAME="E12:44"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> 
 ;<br/>

<b>consider </b><font color="Maroon">T</font> being  <a href="glib_001.html#V5">Path-like</a>  <a href="glib_001.html#M4">Subwalk</a> of <font color="Maroon">W2</font>;<br/>
<a NAME="E14:44"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="glib_001.html#T163">GLIB_001:163</a></i>;<br/>
<a NAME="E15:44"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K4">.last()</a>  )
 
<b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E16:44"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  )
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E17:44"/>
( <font color="Maroon">T</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">T</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K4">.last()</a>  )
 
<b>by </b><i><a class="ref" href="glib_001.html#T162">GLIB_001:162</a></i>;<br/>
<a NAME="E18:44"/><b>then </b>
<font color="Maroon">T</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 
<b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="chord.html#T7" target="_self">Th7</a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E19:44"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E20:44"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E21:44"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div>
