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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>assume </b><a NAME="E1:43"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">P2</font> being  <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>  st <font color="Olive">P2</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>  holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">P2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">V</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">V</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/><a NAME="E2:43_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">V</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">V</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/><b>consider </b><font color="Maroon">P3</font> being  <a href="glib_001.html#V5">Path-like</a>  <a href="glib_001.html#M4">Subwalk</a> of <font color="Maroon">V</font>;<br/><a NAME="E4:43_1"/>
( <font color="Maroon">P3</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">P3</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="chord.html#T3" target="_self">Th3</a>, <a class="ref" href="glib_001.html#T162">GLIB_001:162</a></i>;<br/><a NAME="E5:43_1"/><b>then </b>
<font color="Maroon">P3</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="E6:43_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P3</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T163">GLIB_001:163</a></i>;<br/><b>hence </b><a NAME="E7:43_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
<font color="Maroon">W</font> is <a href="chord.html#V1" target="_self">minlength</a>
 <b>by </b><i/>;<br/>


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