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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/> for <font color="Olive" title="b3">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b4">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds  not <font color="Olive" title="b4">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</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="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  for <font color="Olive" title="b1">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds  not <font color="Olive" title="b2">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E2:83"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
 not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</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">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds  not <font color="Olive" title="b2">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">S</font> be    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds  not <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:83"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds  not <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E4:83"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</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">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T34" onmouseover="rs('glib_000/T34')" onmouseout="rh()">GLIB_000:34</a></span>;<br/>
<a NAME="E5:83"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">G2</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font>;<br/>
<span class="kw">given </span><font color="Maroon" title="c6">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><a NAME="E7:83"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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/>

<a NAME="E8:83"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font> <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="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E9:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font>
 
;<br/>
<a NAME="E10:83"/>
 the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="glib_000.html#T34" onmouseover="rs('glib_000/T34')" onmouseout="rh()">GLIB_000:34</a></span>;<br/>
<a NAME="E11:83"/><span class="kw">then </span>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font> <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="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E12:83"/><span class="kw">then </span>
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">W2</font> = <font color="Maroon" title="c6">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="glib_001.html#T170" onmouseover="rs('glib_001/T170')" onmouseout="rh()">GLIB_001:170</a></span>;<br/>
<a NAME="E14:83"/>
<font color="Maroon" title="c6">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="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E75">A6</font></span></a></span>;<br/>
<a NAME="E15:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A8</font></span>: 
<font color="Maroon" title="c7">W2</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">b</font>
 
;<br/>
<a NAME="E16:83"/>
<font color="Maroon" title="c6">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="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E75">A6</font></span></a></span>;<br/>
<a NAME="E17:83"/><span class="kw">then </span>
<font color="Maroon" title="c7">W2</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">a</font>
 
;<br/>
<a NAME="E18:83"/><span class="kw">then </span>
<font color="Maroon" title="c7">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83"><span class="lab"><font color="Green" title="E77">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:83"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" onmouseover="rs('chord/D8')" onmouseout="rh()">Def8</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/>


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