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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">  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/><span class="kw">let </span><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" 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> <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><span class="kw">hereby </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"> ( <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"><span class="kw">assume </span><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<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" 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">u</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1_1"><span class="kw">now </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">  not <font color="Maroon" title="c2">u</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_1"/>
<font color="Maroon" title="c2">u</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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" 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="c3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:61_1_1"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<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#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <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">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:61_1_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/><a NAME="E6:61_1_1"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="glib_000.html#D18" onmouseover="rs('glib_000/D18')" onmouseout="rh()">GLIB_000:def 18</a></span>;<br/><a NAME="E7:61_1_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" onmouseover="rs('chord/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:61_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E6:61_1_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</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><span class="kw">hence </span><a NAME="E3:61_1"/>
<font color="Maroon" title="c2">u</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</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>

<span class="kw">assume </span><a NAME="E2:61"/>
<font color="Maroon" title="c2">u</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</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> <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="E3:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">now </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> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">  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/><span class="kw">assume </span><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<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" 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">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E52">A6</font></span></a></span>;<br/><a NAME="E3:61_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E4:61_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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>
 ;<br/><span class="kw">hence </span><a NAME="E5:61_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a>, <a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</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>
<span class="kw">hence </span><a NAME="E5:61"/>
<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 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>; <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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