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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">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b2">G2</font> being   <a href="glib_007.html#NM8" title="GLIB_007:NM.8">addAdjVertexAll</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">v</font><br/>  for <font color="Olive" title="b3">G3</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b4">G4</font> being   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Olive" title="b3">G3</font>,<font color="Olive" title="b1">v</font>  st  not <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="b2">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="Olive" title="b3">G3</font> holds <br/><font color="Olive" title="b4">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Olive" title="b2">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">G2</font> being   <a href="glib_007.html#NM8" title="GLIB_007:NM.8">addAdjVertexAll</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">v</font><br/>  for <font color="Olive" title="b2">G3</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b3">G4</font> being   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Olive" title="b2">G3</font>,<font color="Maroon" title="c2">v</font>  st  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b2">G3</font> holds <br/><font color="Olive" title="b3">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Olive" title="b1">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">G2</font> be   <a href="glib_007.html#NM8" title="GLIB_007:NM.8">addAdjVertexAll</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">v</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">G3</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">G4</font> being   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Olive" title="b1">G3</font>,<font color="Maroon" title="c2">v</font>  st  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">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="Olive" title="b1">G3</font> holds <br/><font color="Olive" title="b2">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c3">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">G3</font> be    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</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">G4</font> being   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Maroon" title="c4">G3</font>,<font color="Maroon" title="c2">v</font>  st  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">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="c4">G3</font> holds <br/><font color="Olive" title="b1">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c3">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">G4</font> be   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Maroon" title="c4">G3</font>,<font color="Maroon" title="c2">v</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"> (  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">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="c4">G3</font> implies <font color="Maroon" title="c5">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c3">G2</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:169"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
(  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">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="c4">G3</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="c5">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c3">G2</font></span><br/>

<a NAME="E2:169"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">G9</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c4">G3</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:169"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
<font color="Maroon" title="c6">G9</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c3">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:169"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_012.html#T105" target="_self" onmouseover="rs('glib_012/T105')" onmouseout="rh()">Th105</a></span>;<br/>
<a NAME="E5:169"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</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_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>;<br/>
<a NAME="E6:169"/><span class="kw">then </span>
<font color="Maroon" title="c6">G9</font> is   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Maroon" title="c4">G3</font>,<font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T55" onmouseover="rs('glib_007/T55')" onmouseout="rh()">GLIB_007:55</a></span>;<br/>
<a NAME="E7:169"/><span class="kw">then </span>
<font color="Maroon" title="c5">G4</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c6">G9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T77" onmouseover="rs('glib_006/T77')" onmouseout="rh()">GLIB_006:77</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:169"/>
<font color="Maroon" title="c5">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c3">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:169"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="glib_012.html#T97" target="_self" onmouseover="rs('glib_012/T97')" onmouseout="rh()">Th97</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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