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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E33">Th31</font></span>: <a NAME="T23"><span class="comment"><font color="firebrick">:: GLPACY00:23</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">P</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  ex <font color="Olive" title="b2">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Olive" title="b1">P</font> st <br/>( <font color="Olive" title="b2">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b2">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b2">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() implies <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() ) &amp; ( <font color="Olive" title="b2">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b2">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</div></div>
