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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-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">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">v</font><br/>  for <font color="Olive" title="b3">C</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Olive" title="b2">F</font><br/>  for <font color="Olive" title="b4">w1</font>, <font color="Olive" title="b5">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">F</font>  st <font color="Olive" title="b4">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">w2</font> &amp; <font color="Olive" title="b4">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b5">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b4">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  holds <br/> not <font color="Olive" title="b5">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</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">  for <font color="Olive" title="b1">F</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">v</font><br/>  for <font color="Olive" title="b2">C</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b3">w1</font>, <font color="Olive" title="b4">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b3">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">w2</font> &amp; <font color="Olive" title="b3">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b4">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  holds <br/> not <font color="Olive" title="b4">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">v</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">  for <font color="Olive" title="b1">C</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c3">F</font><br/>  for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">F</font>  st <font color="Olive" title="b2">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w2</font> &amp; <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  holds <br/> not <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">C</font> be   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c3">F</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">  for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">F</font>  st <font color="Olive" title="b1">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w2</font> &amp; <font color="Olive" title="b1">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b2">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  holds <br/> not <font color="Olive" title="b2">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">w1</font>, <font color="Maroon" title="c6">w2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">F</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="c5">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w2</font> &amp; <font color="Maroon" title="c5">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c6">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  implies  not <font color="Maroon" title="c6">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c5">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w2</font>
 <span class="kw">and </span><br/><a NAME="E2:5"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
( <font color="Maroon" title="c5">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c6">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">C</font> )
 <span class="kw">and </span><br/><a NAME="E3:5"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
( <font color="Maroon" title="c5">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  &amp; <font color="Maroon" title="c6">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</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"> contradiction</span><br/>

<a NAME="E4:5"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c5">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="glib_002.html#T33" onmouseover="rs('glib_002/T33')" onmouseout="rh()">GLIB_002:33</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">W9</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">F</font><span class="kw"> such that </span><br/><a NAME="E6:5"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c7">W9</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c6">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="glib_002.html#D5" onmouseover="rs('glib_002/D5')" onmouseout="rh()">GLIB_002:def 5</a></span>;<br/>
<a NAME="E7:5"/>
(  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>;<br/>
<a NAME="E8:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 not <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">W9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">W</font> = <font color="Maroon" title="c7">W9</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" onmouseover="rs('glib_001/T167')" onmouseout="rh()">GLIB_001:167</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">P8</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E11:5"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="glib_000.html#T71" onmouseover="rs('glib_000/T71')" onmouseout="rh()">GLIB_000:71</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:5"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c11">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">w2</font>,<font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="glib_000.html#T71" onmouseover="rs('glib_000/T71')" onmouseout="rh()">GLIB_000:71</a></span>;<br/>
<a NAME="E14:5"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c11">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">w2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:5"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">P9</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">P</font> = <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c10">e1</font>;<br/>
<a NAME="E15:5"/>
<font color="Maroon" title="c8">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c6">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</a></span>;<br/>
<a NAME="E16:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A10</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c6">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T160" onmouseover="rs('glib_001/T160')" onmouseout="rh()">GLIB_001:160</a></span>;<br/>
<a NAME="E17:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<font color="Maroon" title="c11">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/>
<a NAME="E18:5"/>
 not <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/>
<a NAME="E19:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A12</font></span>: 
 not <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T163" onmouseover="rs('glib_001/T163')" onmouseout="rh()">GLIB_001:163</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E20:5"/>
(  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w1</font> &amp;  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</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="c6">w2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/>
<a NAME="E21:5"/><span class="kw">then </span>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E5">A1</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/>
<a NAME="E22:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A13</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:5"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E19:5"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="glib_001.html#T151" onmouseover="rs('glib_001/T151')" onmouseout="rh()">GLIB_001:151</a></span>;<br/>
<a NAME="E23:5"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="glib_001.html#T66" onmouseover="rs('glib_001/T66')" onmouseout="rh()">GLIB_001:66</a></span>;<br/>
<a NAME="E24:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <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="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/>
<a NAME="E25:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A17</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c10">e1</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="glib_001.html#T132" onmouseover="rs('glib_001/T132')" onmouseout="rh()">GLIB_001:132</a></span>;<br/>
<a NAME="E26:5"/><span class="lab"><font color="Green" title="E20">A18</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c10">e1</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E23:5"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="glib_001.html#T66" onmouseover="rs('glib_001/T66')" onmouseout="rh()">GLIB_001:66</a>, <a class="ref" href="glib_001.html#T119" onmouseover="rs('glib_001/T119')" onmouseout="rh()">GLIB_001:119</a></span>;<br/>
<a NAME="E27:5"/><span class="lab"><font color="Green" title="E21">A19</font></span>: 
 not <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:5_1"/>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</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"> contradiction</span><br/>

<a NAME="E2:5_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">e2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:5"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="glib_001.html#T111" onmouseover="rs('glib_001/T111')" onmouseout="rh()">GLIB_001:111</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_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="E1:5_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  or <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T136" onmouseover="rs('zfmisc_1/T136')" onmouseout="rh()">ZFMISC_1:136</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_1"/>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</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"> contradiction</span><br/><div class="add"><a NAME="E2:5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T163" onmouseover="rs('glib_001/T163')" onmouseout="rh()">GLIB_001:163</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E3:5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="glib_001.html#T105" onmouseover="rs('glib_001/T105')" onmouseout="rh()">GLIB_001:105</a></span>;<br/><span class="kw">hence </span><a NAME="E4:5_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>; <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="5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_2"/>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e2</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"> contradiction</span><br/><div class="add"><a NAME="E2:5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">W9</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E13:5"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a></span>;<br/><a NAME="E3:5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">W9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:5_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>; <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>

</div><span class="kw">end;</span></div>
<a NAME="E28:5"/>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E24:5"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</a></span>;<br/>
<a NAME="E29:5"/><span class="kw">then </span>
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c10">e1</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:5"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E27:5"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="ref" href="glib_002.html#T42" onmouseover="rs('glib_002/T42')" onmouseout="rh()">GLIB_002:42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:5"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="txt" href="#E26:5"><span class="lab"><font color="Green" title="E20">A18</font></span></a></span>; <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>
