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<div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">P</font> =  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:99_1_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">v</font> =  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>;<br/><span class="kw">take </span><font color="Maroon" title="c4">C</font> =  the    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>, the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c4">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )</span><br/><a NAME="E2:99_1_1_1_1"/>
<font color="Maroon" title="c4">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1_1_1_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="glpacy00.html#T33" target="_self" onmouseover="rs('glpacy00/T33')" onmouseout="rh()">Th41</a></span>;<br/><span class="kw">hence </span><a NAME="E3:99_1_1_1_1"/>
( <font color="Maroon" title="c4">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c4">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_1_1_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Olive">b<sub>1</sub></font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )</span><br/><div class="add"><a NAME="E2:99_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:99_1_1_1_2"/><span class="kw">then </span>
 the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>;<br/><a NAME="E4:99_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 not  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c3">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  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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><span class="kw"> such that </span><br/><a NAME="E6:99_1_1_1_2"/>
( <font color="Maroon" title="c3">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>  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> &amp; <font color="Maroon" title="c3">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>  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> )
 <span class="kw">and </span><br/><a NAME="E7:99_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
(  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> <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="Maroon" title="c3">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="Maroon" title="c3">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> iff  not  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E8:99_1_1_1_2"/>
( <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() iff  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E9:99_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  iff  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E10:99_1_1_1_2"/>
<font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T23" target="_self" onmouseover="rs('glpacy00/T23')" onmouseout="rh()">Th31</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> , <font color="Maroon" title="c5">v2</font> = <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_1_1_1_2"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E7:99_1_1_1_2"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E12:99_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_1_1_1_2"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E9:99_1_1_1_2"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">C</font> =  the    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>,<font color="Maroon" title="c4">v1</font>, <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a>  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>,<font color="Maroon" title="c5">v2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c6">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c6">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )</span><br/><a NAME="E13:99_1_1_1_2"/>
 not  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a>  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a>  the  <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</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>
 ;<br/><a NAME="E14:99_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_1_1_1_2"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E12:99_1_1_1_2"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="glpacy00.html#T34" target="_self" onmouseover="rs('glpacy00/T34')" onmouseout="rh()">Th42</a></span>;<br/><span class="kw">hence </span><a NAME="E15:99_1_1_1_2"/>
( <font color="Maroon" title="c6">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c6">C</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c6">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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