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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</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="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  iff  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</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">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</font> holds <br/> not <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:66"/>
( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</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">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</font> holds <br/> not <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T59" target="_self" onmouseover="rs('glibpre0/T59')" onmouseout="rh()">Th65</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"> ( (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</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">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</font> holds <br/> not <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:66"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</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">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</font> holds <br/> not <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</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"> <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:66"/>
 not <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:66"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">v</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">w</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>, <font color="Maroon" title="c6">w</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<a NAME="E7:66"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<a NAME="E8:66"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:66"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</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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