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<span class="kw">let </span><font color="Maroon" title="c1">G</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"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V5" title="GLIB_012:attr.5">self-Dcomplementary</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V6" title="GLIB_012:attr.6">self-complementary</a>  implies ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp; <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="E1:193_2_1"/><span class="lab"><font color="Green" title="E89">A1</font></span>: 
( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V5" title="GLIB_012:attr.5">self-Dcomplementary</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V6" title="GLIB_012:attr.6">self-complementary</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_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp; <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">set </span><font color="Maroon" title="c2">H1</font> =  the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">H2</font> =  the    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E2:193_2_1"/>
 the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font> is <font color="Maroon" title="c1">G</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_2_1"><span class="lab"><font color="Green" title="E89">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">F1</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G</font>, the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:193_2_1"/><span class="lab"><font color="Green" title="E90">A2</font></span>: 
<font color="Maroon" title="c4">F1</font> is  <a href="glib_010.html#V18" title="GLIB_010:attr.18">Disomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#D24" onmouseover="rs('glib_010/D24')" onmouseout="rh()">GLIB_010:def 24</a></span>;<br/>
<a NAME="E5:193_2_1"/>
 the    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G</font> is <font color="Maroon" title="c1">G</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_2_1"><span class="lab"><font color="Green" title="E89">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">F2</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G</font>, the    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E7:193_2_1"/><span class="lab"><font color="Green" title="E91">A3</font></span>: 
<font color="Maroon" title="c5">F2</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#D23" onmouseover="rs('glib_010/D23')" onmouseout="rh()">GLIB_010:def 23</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="193_2_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">  for <font color="Olive" title="b1">v1</font>, <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="c1">G</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> holds <br/><font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</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="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> implies <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:193_2_1_1"/><span class="lab"><font color="Green" title="E92">A4</font></span>: 
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">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="c6">v1</font>,<font color="Maroon" title="c7">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/><span class="kw">assume </span><a NAME="E2:193_2_1_1"/><span class="lab"><font color="Green" title="E93">A5</font></span>: 
 not <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</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"> contradiction</span><br/><a NAME="E3:193_2_1_1"/>
 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">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="193_2_1_1_1"><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="E2:193_2_1_1_1"/><span class="lab"><font color="Green" title="E94">A6</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">v1</font>,<font color="Maroon" title="c7">v2</font>,<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"> contradiction</span><br/>

<a NAME="E3:193_2_1_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:193_2_1_1_1"><span class="lab"><font color="Green" title="E94">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="E4:193_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:193_2_1_1"><span class="lab"><font color="Green" title="E93">A5</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">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:193_2_1_1"/><span class="lab"><font color="Green" title="E94">A7</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>, the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_2_1_1"><span class="lab"><font color="Green" title="E92">A4</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="E6:193_2_1_1"/>
 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="c7">v2</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="193_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c9">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:193_2_1_1_2"/><span class="lab"><font color="Green" title="E95">A8</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c6">v1</font>,<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"> contradiction</span><br/>

<a NAME="E3:193_2_1_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:193_2_1_1_2"><span class="lab"><font color="Green" title="E95">A8</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="E4:193_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:193_2_1_1"><span class="lab"><font color="Green" title="E93">A5</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="c9">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:193_2_1_1"/><span class="lab"><font color="Green" title="E95">A9</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c6">v1</font>, the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_2_1_1"><span class="lab"><font color="Green" title="E92">A4</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="E9:193_2_1_1"/><span class="lab"><font color="Green" title="E96">A10</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>, the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>, the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:193_2_1_1"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E8:193_2_1_1"><span class="lab"><font color="Green" title="E95">A9</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="E10:193_2_1_1"/>
 the    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_2_1"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E4:193_2_1"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="glib_010.html#T89" onmouseover="rs('glib_010/T89')" onmouseout="rh()">GLIB_010:89</a></span>;<br/><a NAME="E11:193_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:193_2_1_1"><span class="lab"><font color="Green" title="E96">A10</font></span></a>, <a class="ref" href="glib_000.html#D20" onmouseover="rs('glib_000/D20')" onmouseout="rh()">GLIB_000:def 20</a></span>;<br/><span class="kw">hence </span><a NAME="E12:193_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193_2_1_1"><span class="lab"><font color="Green" title="E92">A4</font></span></a>, <a class="txt" href="#E5:193_2_1_1"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E8:193_2_1_1"><span class="lab"><font color="Green" title="E95">A9</font></span></a>, <a class="ref" href="glib_000.html#T125" onmouseover="rs('glib_000/T125')" onmouseout="rh()">GLIB_000:125</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>
<a NAME="E9:193_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">G</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D6" onmouseover="rs('chord/D6')" onmouseout="rh()">CHORD:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:193_2_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:193_2_1"><span class="lab"><font color="Green" title="E91">A3</font></span></a>, <a class="ref" href="glib_010.html#T89" onmouseover="rs('glib_010/T89')" onmouseout="rh()">GLIB_010:89</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>
