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<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Olive" title="b1">u</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> )</span><br/><b>let </b><font color="Maroon" title="c2">u</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c2">u</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">u</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> implies <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span>
<div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">u</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1"/>
<font color="Maroon" title="c2">u</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:63_1_1"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:63_1_1"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:63_1_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="glib_000.html#T67" title="GLIB_000:th.67">GLIB_000:67</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_1"/>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:63_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">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> &amp; <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="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <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="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> )
 <b>by </b><i><a class="txt" href="chord.html#E5:63_1_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><b>hence </b><a NAME="E3:63_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_000.html#D20" title="GLIB_000:def.20">GLIB_000:def 20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E7:63_1_1"/><b>then </b>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">u</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E5:63_1_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E8:63_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:63_1"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:63_1"/>
<font color="Maroon" title="c2">u</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D51" title="GLIB_000:def.51">GLIB_000:def 51</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:63"/>
<font color="Maroon" title="c2">u</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E3:63"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="glib_000.html#D51" title="GLIB_000:def.51">GLIB_000:def 51</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>assume </b><a NAME="E1:63_2"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>reconsider </b><font color="Maroon" title="c4">v</font> = <font color="Maroon" title="c3">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E1:63_2"><i><font color="Green" title="E51">A5</font></i></a></i>;<br/><a NAME="E3:63_2"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">u</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:63_2"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:63_2"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><b>thus </b><a NAME="E6:63_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="chord.html#E5:63_2"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a>, <a class="ref" href="glib_000.html#T65" title="GLIB_000:th.65">GLIB_000:65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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