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<span class="kw">let </span><font color="Maroon" title="c1">G1</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">G2</font> being    <a href="glib_012.html#M2" title="GLIB_012:mode.2">DLGraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b3">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G2</font>  st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> holds <br/>( ( <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies  not <font color="Olive" title="b3">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies  not <font color="Olive" title="b3">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_012.html#M2" title="GLIB_012:mode.2">DLGraphComplement</a> of <font color="Maroon" title="c1">G1</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">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> holds <br/>( ( <font color="Olive" title="b1">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies  not <font color="Olive" title="b2">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Olive" title="b1">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies  not <font color="Olive" title="b2">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</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">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> holds <br/>( ( <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies  not <font color="Olive" title="b1">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies  not <font color="Olive" title="b1">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</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="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> implies ( ( <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:102"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</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="c3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</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="c3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:102_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c3">v1</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">  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/><a NAME="E2:102_1"/>
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="102_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c5">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:102_1_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</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="E3:102_1_1"/>
<font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:102_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="glib_000.html#T143" onmouseover="rs('glib_000/T143')" onmouseout="rh()">GLIB_000:143</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">then </span><span class="kw">consider </span><font color="Maroon" title="c5">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:102_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E5:102_1"/>
<font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E4:102_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">hence </span><a NAME="E6:102_1"/>
 not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T143" onmouseover="rs('glib_000/T143')" onmouseout="rh()">GLIB_000:143</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">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"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:102_2"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</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="c4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> </span><br/><a NAME="E2:102_2"/>
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="102_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c5">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:102_2_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</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="E3:102_2_1"/>
<font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_2_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:102_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_2"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="glib_000.html#T146" onmouseover="rs('glib_000/T146')" onmouseout="rh()">GLIB_000:146</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">then </span><span class="kw">consider </span><font color="Maroon" title="c5">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:102_2"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E5:102_2"/>
<font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E4:102_2"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">hence </span><a NAME="E6:102_2"/>
 not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T146" onmouseover="rs('glib_000/T146')" onmouseout="rh()">GLIB_000:146</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>
