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<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_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> holds <br/> ( <font color="Olive" title="b1">G2</font> is    <a href="glib_012.html#M3" title="GLIB_012:mode.3">LGraphComplement</a> of <font color="Maroon" title="c1">G1</font> iff (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Olive" title="b1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b1">G2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> &amp; (  for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">w1</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="b4">v2</font>, <font color="Olive" title="b5">w2</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="b4">v2</font> &amp; <font color="Olive" title="b3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w2</font> holds <br/>( <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  iff  not <font color="Olive" title="b4">v2</font>,<font color="Olive" title="b5">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</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"> ( <font color="Maroon" title="c2">G2</font> is    <a href="glib_012.html#M3" title="GLIB_012:mode.3">LGraphComplement</a> of <font color="Maroon" title="c1">G1</font> iff (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">w1</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>, <font color="Olive" title="b4">w2</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="b3">v2</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w2</font> holds <br/>( <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  iff  not <font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</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"> (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">w1</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>, <font color="Olive" title="b4">w2</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="b3">v2</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w2</font> holds <br/>( <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  iff  not <font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) ) implies <font color="Maroon" title="c2">G2</font> is    <a href="glib_012.html#M3" title="GLIB_012:mode.3">LGraphComplement</a> of <font color="Maroon" title="c1">G1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:109_1"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c2">G2</font> is    <a href="glib_012.html#M3" title="GLIB_012:mode.3">LGraphComplement</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"> (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">w1</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>, <font color="Olive" title="b4">w2</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="b3">v2</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w2</font> holds <br/>( ( <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  not <font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) &amp; (  not <font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) ) ) )</span><br/><span class="kw">hence </span><a NAME="E2:109_1"/>
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D7" target="_self" onmouseover="rs('glib_012/D7')" onmouseout="rh()">Def7</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">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">w1</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>, <font color="Olive" title="b4">w2</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="b3">v2</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w2</font> holds <br/>( ( <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  not <font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) &amp; (  not <font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">w1</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>, <font color="Olive" title="b2">w2</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; <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w2</font> holds <br/>( ( <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  not <font color="Olive" title="b1">v2</font>,<font color="Olive" title="b2">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) &amp; (  not <font color="Olive" title="b1">v2</font>,<font color="Olive" title="b2">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">v2</font>, <font color="Maroon" title="c6">w2</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="c5">v2</font> &amp; <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w2</font> implies ( ( <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) &amp; (  not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) ) )</span><br/><span class="kw">assume </span><a NAME="E3:109_1"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> &amp; <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w2</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>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) &amp; (  not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</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"> (  not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:109_1_1"/>
<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</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">  not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/><a NAME="E2:109_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</a></span>;<br/><a NAME="E3:109_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_1"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="glib_012.html#D7" target="_self" onmouseover="rs('glib_012/D7')" onmouseout="rh()">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:109_1_1"/>
 not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</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="#E3:109_1"><span class="lab"><font color="Green" title="E49">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">assume </span><a NAME="E5:109_1"/>
 not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</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"> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/><a NAME="E6:109_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</a></span>;<br/><a NAME="E7:109_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_1"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="glib_012.html#D7" target="_self" onmouseover="rs('glib_012/D7')" onmouseout="rh()">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:109_1"/>
<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><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">assume </span><span class="kw">that </span><br/><a NAME="E2:109"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> )
 <span class="kw">and </span><br/><a NAME="E3:109"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
 for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">w1</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>, <font color="Olive" title="b4">w2</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="b3">v2</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w2</font> holds <br/>( <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  iff  not <font color="Olive" title="b3">v2</font>,<font color="Olive" title="b4">w2</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"> <font color="Maroon" title="c2">G2</font> is    <a href="glib_012.html#M3" title="GLIB_012:mode.3">LGraphComplement</a> of <font color="Maroon" title="c1">G1</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="109_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">v1</font>, <font color="Olive" title="b2">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( (  ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font>,<font color="Maroon" title="c1">G1</font> implies  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font>,<font color="Maroon" title="c2">G2</font> ) implies  ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font>,<font color="Maroon" title="c1">G1</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">w1</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"> ( (  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</font>,<font color="Maroon" title="c1">G1</font> implies  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<font color="Maroon" title="c2">G2</font> ) implies  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</font>,<font color="Maroon" title="c1">G1</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">v2</font> = <font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c6">w2</font> = <font color="Maroon" title="c4">w1</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="txt" href="#E2:109"><span class="lab"><font color="Green" title="E48">A3</font></span></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"> ( (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<font color="Maroon" title="c2">G2</font> ) implies  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</font>,<font color="Maroon" title="c1">G1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:109_2_1"/>
 ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</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">  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<font color="Maroon" title="c2">G2</font></span><br/><a NAME="E2:109_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</a></span>;<br/><a NAME="E3:109_2_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</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="#E3:109"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:109_2_1"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><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">assume </span><a NAME="E3:109_2"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</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="c4">w1</font>,<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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</font>,<font color="Maroon" title="c1">G1</font></span><br/><a NAME="E4:109_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</a></span>;<br/><a NAME="E5:109_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</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="#E3:109"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:109_2"/>
 ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">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="c4">w1</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><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">hence </span><a NAME="E5:109"/>
<font color="Maroon" title="c2">G2</font> is    <a href="glib_012.html#M3" title="GLIB_012:mode.3">LGraphComplement</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="glib_012.html#D7" target="_self" onmouseover="rs('glib_012/D7')" onmouseout="rh()">Def7</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>
