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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="helly.html#NM1" title="HELLY:NM.1">_Tree</a>;<br/><b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green" title="E37">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
( <font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> or <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> or ( not <font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; not <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/>
<font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_1"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="helly.html#E1:45"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="lexbfs.html#T9" title="LEXBFS:th.9">LEXBFS:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/>
<font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_2"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T131" title="GLIB_001:th.131">GLIB_001:131</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_3"><b>suppose </b></a><a NAME="E1:45_1_3"/>
( not <font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; not <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> )
 ;<br/><div class="add"><a NAME="E2:45_1_3"/><b>then </b>
( not <font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> )
 <b>by </b><i><a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a></i>;<br/><b>hence </b><a NAME="E3:45_1_3"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="helly.html#E1:45"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="helly.html#E2:45"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="helly.html#T18" target="_self" title="HELLY:th.18">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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