<?xml version="1.0"?>
<div class="add">

<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#M4" title="GLIB_012:mode.4">DGraphComplement</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> &amp; 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  holds <br/> 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#M4" title="GLIB_012:mode.4">DGraphComplement</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> &amp; 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Olive" title="b1">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  holds <br/> 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> &amp; 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  holds <br/> 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> &amp; 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</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="E1:154"/><span class="lab"><font color="Green" title="E71">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> &amp; 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</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">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or  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="E2:154"/>
<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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:154"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</font> &amp; <font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c3">v1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c6">w</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="#E1:154"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="glibpre0.html#T95" onmouseover="rs('glibpre0/T95')" onmouseout="rh()">GLIBPRE0:95</a></span>;<br/>
<a NAME="E5:154"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</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="c6">w</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="154_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c7">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:154_1"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
<font color="Maroon" title="c7">e</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="c6">w</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="E3:154_1"/>
<font color="Maroon" title="c7">e</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="c6">w</font>,<font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:154_1"><span class="lab"><font color="Green" title="E73">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="E4:154_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:154"><span class="lab"><font color="Green" title="E72">A2</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:154"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
<font color="Maroon" title="c7">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="c6">w</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:154"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="glib_012.html#T80" target="_self" onmouseover="rs('glib_012/T80')" onmouseout="rh()">Th80</a></span>;<br/>
<a NAME="E8:154"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">w</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="154_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:154_2"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">w</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="E3:154_2"/>
<font color="Maroon" title="c8">e</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="c6">w</font>,<font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:154_2"><span class="lab"><font color="Green" title="E74">A5</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="E4:154_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:154"><span class="lab"><font color="Green" title="E72">A2</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:154"/><span class="lab"><font color="Green" title="E74">A6</font></span>: 
<font color="Maroon" title="c8">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">w</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="txt" href="#E4:154"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="glib_012.html#T80" target="_self" onmouseover="rs('glib_012/T80')" onmouseout="rh()">Th80</a></span>;<br/>
<a NAME="E11:154"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
<font color="Maroon" title="c7">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:154"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E7:154"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E10:154"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="glib_000.html#T125" onmouseover="rs('glib_000/T125')" onmouseout="rh()">GLIB_000:125</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">w</font> = <font color="Maroon" title="c6">w</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T80" target="_self" onmouseover="rs('glib_012/T80')" onmouseout="rh()">Th80</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">e1</font> = <font color="Maroon" title="c7">e1</font>, <font color="Maroon" title="c11">e2</font> = <font color="Maroon" title="c8">e2</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E14:154"/>
( <font color="Maroon" title="c10">e1</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="c9">w</font>,<font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c11">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="c9">w</font>,<font color="Maroon" title="c2">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:154"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E7:154"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E10:154"><span class="lab"><font color="Green" title="E74">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/>
<a NAME="E15:154"/><span class="kw">then </span>
( <font color="Maroon" title="c10">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  &amp; <font color="Maroon" title="c11">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">v2</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="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/>
<a NAME="E16:154"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A8</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">e1</font>,<font color="Maroon" title="c11">e2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">v2</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="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a></span>;<br/>
<span class="kw">assume </span><a NAME="E17:154"/>
<font color="Maroon" title="c4">v2</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">e9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E19:154"/><span class="lab"><font color="Green" title="E77">A9</font></span>: 
( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">e9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c12">e9</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="ref" href="glib_000.html#D51" onmouseover="rs('glib_000/D51')" onmouseout="rh()">GLIB_000:def 51</a></span>;<br/>
<a NAME="E20:154"/>
( <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e9</font> &amp; <font color="Maroon" title="c11">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:154"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E19:154"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T20" onmouseover="rs('zfmisc_1/T20')" onmouseout="rh()">ZFMISC_1:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:154"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:154"><span class="lab"><font color="Green" title="E75">A7</font></span></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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