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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">V</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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><br/>  for <font color="Olive" title="b3">G2</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b4">G3</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</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="b3">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="b4">G3</font> holds <br/> ex <font color="Olive" title="b5">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Olive" title="b4">G3</font>,<font color="Olive" title="b1">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> <font color="Olive" title="b2">V</font> st <font color="Olive" title="b5">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Olive" title="b3">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">V</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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><br/>  for <font color="Olive" title="b2">G2</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">v</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>  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="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/> ex <font color="Olive" title="b4">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Olive" title="b3">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> <font color="Olive" title="b1">V</font> st <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="c3">V</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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 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#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">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>  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/> ex <font color="Olive" title="b3">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Olive" title="b2">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> <font color="Maroon" title="c3">V</font> st <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="c4">G2</font> be    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">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>  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="c4">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/> ex <font color="Olive" title="b2">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Olive" title="b1">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> <font color="Maroon" title="c3">V</font> st <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="c4">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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"> (  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="c4">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="c5">G3</font> implies  ex <font color="Olive" title="b1">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">V</font> st <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="c4">G2</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:167"/><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="c4">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="c5">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">  ex <font color="Olive" title="b1">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">V</font> st <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="c4">G2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">E0</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:167"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">E0</font> &amp; <font color="Maroon" title="c6">E0</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="E4:167"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">E0</font>
 <span class="kw">and </span><br/><a NAME="E5:167"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
 for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E0</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1"/>
( <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> <font color="Maroon" title="c3">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <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> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_1"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<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> <font color="Maroon" title="c3">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  ex <font color="Olive" title="b1">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">V</font> st <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="c4">G2</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c7">E</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  = $1 <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> ;<br/><span class="kw">consider </span><font color="Maroon" title="c8">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:167_1_1"/><span class="lab"><font color="Green" title="E85">A4</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  holds <br/><font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" onmouseover="rs('funct_1/S3')" onmouseout="rh()">FUNCT_1:sch 3</a>();<br/></span><span class="kw">set </span><font color="Maroon" title="c9">s</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">h</font>;<br/><span class="lab"><font color="Green" title="E86">A5</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_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">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font> ) &amp; ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font> implies <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">y</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"> ( ( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font> ) &amp; ( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font> implies <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> ) )</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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font> implies <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> )</span>
<div class="add"><span class="kw">set </span><font color="Maroon" title="c11">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><span class="kw">assume </span><a NAME="E1:167_1_1_2_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font></span><br/><a NAME="E2:167_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 ;<br/><span class="kw">then </span><font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">y</font>

;<br/><span class="kw">hence </span><a NAME="E4:167_1_1_2_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E2:167_1_1_2_1"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1: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="E2:167_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:167_1_1_2"/><span class="lab"><font color="Green" title="E87">A7</font></span>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</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/><span class="kw">consider </span><font color="Maroon" title="c13">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E7:167_1_1_2"/><span class="lab"><font color="Green" title="E88">A8</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c13">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E4:167_1_1_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a></span>;<br/><font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
<font color="Maroon" title="c12">x</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E4:167_1_1_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:167_1_1_2"><span class="lab"><font color="Green" title="E88">A8</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E9:167_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E4:167_1_1_2"><span class="lab"><font color="Green" title="E87">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/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A9</font></span>:  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</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> <font color="Maroon" title="c3">V</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>
<br/>.= 
<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="c5">G3</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>
;<br/><a NAME="E7:167_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A10</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font></span>)</span> <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="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><a NAME="E8:167_1_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T17" onmouseover="rs('funct_4/T17')" onmouseout="rh()">FUNCT_4:17</a></span>;<br/><a NAME="E9:167_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A11</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">h</font></span>)</span> <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="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:167_1_1"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E10:167_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">s</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E9:167_1_1"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">t</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E90">A12</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/><a NAME="E13:167_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E14:167_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A13</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E15:167_1_1"/><span class="lab"><font color="Green" title="E92">A14</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E16:167_1_1"/><span class="lab"><font color="Green" title="E93">A15</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T17" onmouseover="rs('funct_4/T17')" onmouseout="rh()">FUNCT_4:17</a></span>;<br/><a NAME="E17:167_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:167_1_1"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E18:167_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:167_1_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">t</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:167_1_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E16:167_1_1"><span class="lab"><font color="Green" title="E93">A15</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/><a NAME="E20:167_1_1"/><span class="lab"><font color="Green" title="E94">A16</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <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="c5">G3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_5"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:167_1_1_5"/><span class="lab"><font color="Green" title="E95">A17</font></span>: 
( <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c13">e</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/>
<span class="kw">consider </span><font color="Maroon" title="c15">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E6:167_1_1_5"/><span class="lab"><font color="Green" title="E96">A18</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c15">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_5"><span class="lab"><font color="Green" title="E95">A17</font></span></a></span>;<br/>
<a NAME="E7:167_1_1_5"/><span class="lab"><font color="Green" title="E97">A19</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E8:167_1_1_5"/><span class="lab"><font color="Green" title="E98">A20</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:167_1_1_5"/>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c15">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_5"><span class="lab"><font color="Green" title="E96">A18</font></span></a>, <a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/>
<a NAME="E10:167_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:167_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_5"><span class="lab"><font color="Green" title="E95">A17</font></span></a>, <a class="txt" href="#E7:167_1_1_5"><span class="lab"><font color="Green" title="E97">A19</font></span></a>, <a class="txt" href="#E8:167_1_1_5"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="ref" href="xregular.html#T8" onmouseover="rs('xregular/T8')" onmouseout="rh()">XREGULAR:8</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="E21:167_1_1"/><span class="lab"><font color="Green" title="E95">A21</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <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">G2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_6"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:167_1_1_6"/><span class="lab"><font color="Green" title="E96">A22</font></span>: 
( <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c13">e</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/>
<span class="kw">consider </span><font color="Maroon" title="c15">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E6:167_1_1_6"/><span class="lab"><font color="Green" title="E97">A23</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c15">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_6"><span class="lab"><font color="Green" title="E96">A22</font></span></a></span>;<br/>
<a NAME="E7:167_1_1_6"/><span class="lab"><font color="Green" title="E98">A24</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E8:167_1_1_6"/><span class="lab"><font color="Green" title="E99">A25</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:167_1_1_6"/>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c15">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_6"><span class="lab"><font color="Green" title="E97">A23</font></span></a>, <a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/>
<a NAME="E10:167_1_1_6"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:167_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_6"><span class="lab"><font color="Green" title="E96">A22</font></span></a>, <a class="txt" href="#E7:167_1_1_6"><span class="lab"><font color="Green" title="E98">A24</font></span></a>, <a class="txt" href="#E8:167_1_1_6"><span class="lab"><font color="Green" title="E99">A25</font></span></a>, <a class="ref" href="xregular.html#T8" onmouseover="rs('xregular/T8')" onmouseout="rh()">XREGULAR:8</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">set </span><font color="Maroon" title="c13">G4</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>);<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_7"><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"> (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</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> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span> &amp; (  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> holds <br/>( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> ) ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:167_1_1_7"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</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> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</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"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span> &amp; (  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> holds <br/>( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E2:167_1_1_7"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p0">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</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">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> holds <br/>( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">e</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c14">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> implies ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> ) )</span><br/><span class="kw">assume </span><a NAME="E3:167_1_1_7"/><span class="lab"><font color="Green" title="E96">A26</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">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"> ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><a NAME="E4:167_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:167_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A27</font></span>: 
 not <font color="Maroon" title="c14">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:167_1_1"><span class="lab"><font color="Green" title="E94">A16</font></span></a>, <a class="txt" href="#E3:167_1_1_7"><span class="lab"><font color="Green" title="E96">A26</font></span></a>, <a class="ref" href="xboole_0.html#T5" onmouseover="rs('xboole_0/T5')" onmouseout="rh()">XBOOLE_0:5</a></span>;<br/><a NAME="E6:167_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A28</font></span>: 
 not <font color="Maroon" title="c14">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span>
 ;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:167_1_1_7"><span class="lab"><font color="Green" title="E97">A27</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</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"> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> = 
<font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_7"><span class="lab"><font color="Green" title="E98">A28</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p0"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">G4</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> </span>)</span>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">t</font>) as    <a href="glib_006.html#M1" title="GLIB_006:mode.1">Supergraph</a> of <font color="Maroon" title="c5">G3</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D9" onmouseover="rs('glib_006/D9')" onmouseout="rh()">GLIB_006:def 9</a></span>;<br/><a NAME="E24:167_1_1"/>
<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> <font color="Maroon" title="c3">V</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/><a NAME="E25:167_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A29</font></span>: 
( <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> <font color="Maroon" title="c3">V</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="c5">G3</font> &amp;  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="c5">G3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_8"><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"> (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> (  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b2">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( (  not <font color="Olive" title="b2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> ) &amp; (  for <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G3</font> ) ) ) ) ) &amp;  ex <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon" title="c3">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b1">E</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="c5">G3</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">E</font> &amp; (  for <font color="Olive" title="b2">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b4">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b4">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">e2</font> ) ) ) ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:167_1_1_8"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> (  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b2">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( (  not <font color="Olive" title="b2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> ) &amp; (  for <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G3</font> ) ) ) ) ) &amp;  ex <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon" title="c3">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b1">E</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="c5">G3</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">E</font> &amp; (  for <font color="Olive" title="b2">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b4">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b4">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">e2</font> ) ) ) ) )</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">  ex <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon" title="c3">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b1">E</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="c5">G3</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">E</font> &amp; (  for <font color="Olive" title="b2">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b4">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b4">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">e2</font> ) ) ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c15">e</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"> (  not <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( (  not <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> ) &amp; (  for <font color="Olive" title="b2">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c5">G3</font> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E1:167_1_1_8_1"/>
 not <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</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="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( (  not <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> ) &amp; (  for <font color="Olive" title="b2">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c5">G3</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_8_1_1"/><span class="lab"><font color="Green" title="E97">A30</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_1_8_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G3</font> or  not <font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_8_1_1_1_1"/>
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">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"> contradiction</span><br/><div class="add"><a NAME="E2:167_1_1_8_1_1_1_1"/><span class="kw">then </span>
<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="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:167_1_1_8_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_8_1_1_1_2"/><span class="lab"><font color="Green" title="E98">A31</font></span>: 
 not <font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">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"> contradiction</span><br/><div class="add"><a NAME="E2:167_1_1_8_1_1_1_2"/>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_8_1_1"><span class="lab"><font color="Green" title="E97">A30</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:167_1_1_8_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 ;<br/><a NAME="E4:167_1_1_8_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A32</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:167_1_1"><span class="lab"><font color="Green" title="E94">A16</font></span></a>, <a class="txt" href="#E1:167_1_1_8_1_1_1_2"><span class="lab"><font color="Green" title="E98">A31</font></span></a>, <a class="ref" href="xboole_0.html#T5" onmouseover="rs('xboole_0/T5')" onmouseout="rh()">XBOOLE_0:5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E6:167_1_1_8_1_1_1_2"/><span class="lab"><font color="Green" title="E100">A33</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c14">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font>

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E4:167_1_1_8_1_1_1_2"><span class="lab"><font color="Green" title="E99">A32</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E4:167_1_1_8_1_1_1_2"><span class="lab"><font color="Green" title="E99">A32</font></span></a>, <a class="txt" href="#E6:167_1_1_8_1_1_1_2"><span class="lab"><font color="Green" title="E100">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">w</font>

;<br/><a NAME="E8:167_1_1_8_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_8_1_1"><span class="lab"><font color="Green" title="E97">A30</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E9:167_1_1_8_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:167_1_1_8_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">v1</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"> ( (  not <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> ) &amp; (  for <font color="Olive" title="b1">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b1">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Olive" title="b1">v2</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Olive" title="b1">v2</font>,<font color="Maroon" title="c5">G3</font> ) )</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">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b1">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Olive" title="b1">v2</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Olive" title="b1">v2</font>,<font color="Maroon" title="c5">G3</font></span>
<div class="add"><span class="kw">assume </span><a NAME="E1:167_1_1_8_1_2"/><span class="lab"><font color="Green" title="E97">A34</font></span>: 
 not <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">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="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font></span><br/><span class="kw">assume </span><a NAME="E2:167_1_1_8_1_2"/><span class="lab"><font color="Green" title="E98">A35</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_1_8_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G3</font> or  not <font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_8_1_2_1_1"/>
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">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"> contradiction</span><br/><div class="add"><a NAME="E2:167_1_1_8_1_2_1_1"/><span class="kw">then </span>
<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="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:167_1_1_8_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_8_1_2_1_2"/><span class="lab"><font color="Green" title="E99">A36</font></span>: 
 not <font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">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"> contradiction</span><br/><div class="add"><a NAME="E2:167_1_1_8_1_2_1_2"/>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_8_1_2"><span class="lab"><font color="Green" title="E98">A35</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:167_1_1_8_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 ;<br/><a NAME="E4:167_1_1_8_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A37</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_8_1_2_1_2"><span class="lab"><font color="Green" title="E99">A36</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E6:167_1_1_8_1_2_1_2"/><span class="lab"><font color="Green" title="E101">A38</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c14">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font>

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E4:167_1_1_8_1_2_1_2"><span class="lab"><font color="Green" title="E100">A37</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E4:167_1_1_8_1_2_1_2"><span class="lab"><font color="Green" title="E100">A37</font></span></a>, <a class="txt" href="#E6:167_1_1_8_1_2_1_2"><span class="lab"><font color="Green" title="E101">A38</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">w</font>

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_2_1_2_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_1_8_1_2_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_8_1_2"><span class="lab"><font color="Green" title="E98">A35</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_2_1_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_8_1_2_1_2_2_1"/>
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"> contradiction</span><br/><div class="add"><a NAME="E2:167_1_1_8_1_2_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:167_1_1_8_1_2_1_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_8_1_2_1_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_8_1_2_1_2_2_2"/>
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v1</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/><div class="add"><span class="kw">hence </span><a NAME="E2:167_1_1_8_1_2_1_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E1:167_1_1_8_1_2"><span class="lab"><font color="Green" title="E97">A34</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c17">v2</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"> ( <font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c17">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c17">v2</font>,<font color="Maroon" title="c14">G4</font> implies <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c17">v2</font>,<font color="Maroon" title="c5">G3</font> )</span><br/><span class="kw">assume </span><a NAME="E3:167_1_1_8_1"/><span class="lab"><font color="Green" title="E97">A39</font></span>: 
( <font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c17">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c17">v2</font>,<font color="Maroon" title="c14">G4</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="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c17">v2</font>,<font color="Maroon" title="c5">G3</font></span><br/><a NAME="E4:167_1_1_8_1"/>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_8_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_8_1_3"/><span class="lab"><font color="Green" title="E98">A40</font></span>: 
 not <font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">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"> contradiction</span><br/>

<a NAME="E2:167_1_1_8_1_3"/>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_8_1"><span class="lab"><font color="Green" title="E97">A39</font></span></a>, <a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</a></span>;<br/>
<a NAME="E3:167_1_1_8_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
;<br/>
<a NAME="E4:167_1_1_8_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A41</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_8_1_3"><span class="lab"><font color="Green" title="E98">A40</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:167_1_1_8_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A42</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span>
 
;<br/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c14">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font> = 
<font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167_1_1_8_1_3"><span class="lab"><font color="Green" title="E100">A42</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1_8_1_3"><span class="lab"><font color="Green" title="E99">A41</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:167_1_1_8_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_8_1"><span class="lab"><font color="Green" title="E97">A39</font></span></a>, <a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</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="E5:167_1_1_8_1"/>
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c17">v2</font>,<font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_8_1"><span class="lab"><font color="Green" title="E97">A39</font></span></a>, <a class="ref" href="glib_006.html#T71" onmouseover="rs('glib_006/T71')" onmouseout="rh()">GLIB_006:71</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">take </span><font color="Maroon" title="c15">E</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </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="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon" title="c3">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Maroon" title="c15">E</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="c5">G3</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</font> &amp; (  for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) ) ) )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_8_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">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">x2</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"> ( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> implies <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:167_1_1_8_2"/><span class="lab"><font color="Green" title="E97">A43</font></span>: 
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</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="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">w1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E3:167_1_1_8_2"/><span class="lab"><font color="Green" title="E98">A44</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c18">w1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">w2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E5:167_1_1_8_2"/><span class="lab"><font color="Green" title="E99">A45</font></span>: 
<font color="Maroon" title="c17">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c19">w2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E1:167_1_1_8_2"><span class="lab"><font color="Green" title="E97">A43</font></span></a></span>;<br/><font color="Maroon" title="c18">w1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c18">w1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E1:167_1_1_8_2"><span class="lab"><font color="Green" title="E97">A43</font></span></a>, <a class="txt" href="#E3:167_1_1_8_2"><span class="lab"><font color="Green" title="E98">A44</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c19">w2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E1:167_1_1_8_2"><span class="lab"><font color="Green" title="E97">A43</font></span></a>, <a class="txt" href="#E5:167_1_1_8_2"><span class="lab"><font color="Green" title="E99">A45</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">w2</font>

;<br/><span class="kw">hence </span><a NAME="E7:167_1_1_8_2"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_8_2"><span class="lab"><font color="Green" title="E98">A44</font></span></a>, <a class="txt" href="#E5:167_1_1_8_2"><span class="lab"><font color="Green" title="E99">A45</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/></div><span class="kw">end;</span></div><a NAME="E4:167_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A46</font></span>: 
<font color="Maroon" title="c8">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><span class="kw">thus </span> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon" title="c3">V</font></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c15">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E6:167_1_1"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="txt" href="#E4:167_1_1_8"><span class="lab"><font color="Green" title="E97">A46</font></span></a>, <a class="ref" href="card_1.html#T70" onmouseover="rs('card_1/T70')" onmouseout="rh()">CARD_1:70</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"> ( <font color="Maroon" title="c15">E</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="c5">G3</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</font> &amp; (  for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E6:167_1_1_8"/>
<font color="Maroon" title="c15">E</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="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:167_1_1"><span class="lab"><font color="Green" title="E94">A16</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"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</font> &amp; (  for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E7:167_1_1_8"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</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="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">v1</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"> ( <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font> implies  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b2">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E8:167_1_1_8"/><span class="lab"><font color="Green" title="E98">A47</font></span>: 
<font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b2">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c17">e1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">v1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">e1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">v1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> ( <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> &amp; <font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font> ) )</span><br/><span class="kw">thus </span><a NAME="E9:167_1_1_8"/><span class="lab"><font color="Green" title="E99">A48</font></span>: 
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:167_1_1_8"><span class="lab"><font color="Green" title="E98">A47</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"> ( <font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font> ) )</span><br/><a NAME="E10:167_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E11:167_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A49</font></span>: 
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font>
 ;<br/><a NAME="E12:167_1_1_8"/><span class="lab"><font color="Green" title="E101">A50</font></span>: 
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:167_1_1_8"><span class="lab"><font color="Green" title="E99">A48</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E102">A51</font></span>: <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c14">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font>

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E9:167_1_1_8"><span class="lab"><font color="Green" title="E99">A48</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c18">e1</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E9:167_1_1_8"><span class="lab"><font color="Green" title="E99">A48</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">v1</font>

;<br/><span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c14">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font> = 
<font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:167_1_1_8"><span class="lab"><font color="Green" title="E101">A50</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:167_1_1_8"><span class="lab"><font color="Green" title="E99">A48</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E15:167_1_1_8"/>
<font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:167_1_1_8"><span class="lab"><font color="Green" title="E100">A49</font></span></a>, <a class="txt" href="#E13:167_1_1_8"><span class="lab"><font color="Green" title="E102">A51</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</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">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> holds <br/><font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c19">e2</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"> ( <font color="Maroon" title="c19">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</font> implies <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e2</font> )</span><br/><span class="kw">assume </span><a NAME="E16:167_1_1_8"/><span class="lab"><font color="Green" title="E103">A52</font></span>: 
<font color="Maroon" title="c19">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c14">G4</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="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e2</font></span><br/><a NAME="E17:167_1_1_8"/>
 not <font color="Maroon" title="c19">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_8_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_8_6"/>
<font color="Maroon" title="c19">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">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"> contradiction</span><br/>

<a NAME="E2:167_1_1_8_6"/><span class="kw">then </span>
<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="c5">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:167_1_1_8_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</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="E18:167_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A53</font></span>: 
 not <font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:167_1_1_8"><span class="lab"><font color="Green" title="E103">A52</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><a NAME="E19:167_1_1_8"/>
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c14">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:167_1_1_8"><span class="lab"><font color="Green" title="E103">A52</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E20:167_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</font>
 ;<br/><a NAME="E21:167_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A54</font></span>: 
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:167_1_1_8"><span class="lab"><font color="Green" title="E104">A53</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E23:167_1_1_8"/><span class="lab"><font color="Green" title="E106">A55</font></span>: 
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c20">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c14">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font>

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E21:167_1_1_8"><span class="lab"><font color="Green" title="E105">A54</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c20">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E21:167_1_1_8"><span class="lab"><font color="Green" title="E105">A54</font></span></a>, <a class="txt" href="#E23:167_1_1_8"><span class="lab"><font color="Green" title="E106">A55</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c20">w</font>

;<br/><a NAME="E25:167_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A56</font></span>: 
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">w</font> or <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:167_1_1_8"><span class="lab"><font color="Green" title="E103">A52</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E26:167_1_1_8"/>
<font color="Maroon" title="c16">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">w</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_8_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_8_8"/>
<font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">w</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="E2:167_1_1_8_8"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E25:167_1_1_8"><span class="lab"><font color="Green" title="E107">A56</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:167_1_1_8_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</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/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E27:167_1_1_8"/>
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:167_1_1_8"><span class="lab"><font color="Green" title="E106">A55</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/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">G4</font> = <font color="Maroon" title="c14">G4</font> as    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:167_1_1"><span class="lab"><font color="Green" title="E96">A29</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c15">G4</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="c15">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c4">G2</font></span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_9"><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"> (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c15">G4</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="c4">G2</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c15">G4</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">G2</font> &amp; (  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c15">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c15">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</font> ) ) ) )</span></a><div class="add"><span class="kw">thus </span> <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c15">G4</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

<br/>.= 
<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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</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"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c15">G4</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">G2</font> &amp; (  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c15">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c15">G4</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">b<sub>5</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">e1</font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c4">G2</font> ) ) ) )</span><br/><a NAME="E2:167_1_1_9"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <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">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E21:167_1_1"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="ref" href="xboole_1.html#T70" onmouseover="rs('xboole_1/T70')" onmouseout="rh()">XBOOLE_1:70</a></span>;<br/><span class="kw">hence </span><a NAME="E3:167_1_1_9"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c15">G4</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">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">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c15">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c15">G4</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">b<sub>5</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">e1</font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c4">G2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">u</font>, <font color="Maroon" title="c17">w</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">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="c16">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">w</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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E4:167_1_1_9"/><span class="lab"><font color="Green" title="E97">A57</font></span>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">w</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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font> ) )</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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font> )</span>
<div class="add"><span class="kw">given </span><font color="Maroon" title="c18">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:167_1_1_9_2"/><span class="lab"><font color="Green" title="E98">A58</font></span>: 
<font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">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">  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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_1_9_2_1"/>
( <font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c1">G1</font> or  not <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_2"><span class="lab"><font color="Green" title="E98">A58</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_9_2_1_1"/><span class="lab"><font color="Green" title="E99">A59</font></span>: 
<font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">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">  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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</font></span><br/><div class="add"><a NAME="E2:167_1_1_9_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">u</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c17">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><a NAME="E3:167_1_1_9_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A60</font></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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1_9"><span class="lab"><font color="Green" title="E97">A57</font></span></a>, <a class="txt" href="#E1:167_1_1_9_2_1_1"><span class="lab"><font color="Green" title="E99">A59</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><a NAME="E4:167_1_1_9_2_1_1"/>
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E1:167_1_1_9_2_1_1"><span class="lab"><font color="Green" title="E99">A59</font></span></a>, <a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:167_1_1_9_2_1_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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:167_1_1"><span class="lab"><font color="Green" title="E96">A29</font></span></a>, <a class="txt" href="#E3:167_1_1_9_2_1_1"><span class="lab"><font color="Green" title="E100">A60</font></span></a>, <a class="ref" href="glib_007.html#T49" onmouseover="rs('glib_007/T49')" onmouseout="rh()">GLIB_007:49</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_9_2_1_2"/><span class="lab"><font color="Green" title="E99">A61</font></span>: 
 not <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</font></span><br/><div class="add"><a NAME="E2:167_1_1_9_2_1_2"/><span class="lab"><font color="Green" title="E100">A62</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#T59" onmouseover="rs('glib_007/T59')" onmouseout="rh()">GLIB_007:59</a></span>;<br/><a NAME="E3:167_1_1_9_2_1_2"/>
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_2"><span class="lab"><font color="Green" title="E98">A58</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E4:167_1_1_9_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_9_2_1_2"><span class="lab"><font color="Green" title="E99">A61</font></span></a>, <a class="txt" href="#E2:167_1_1_9_2_1_2"><span class="lab"><font color="Green" title="E100">A62</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:167_1_1_9_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D30" onmouseover="rs('glib_000/D30')" onmouseout="rh()">GLIB_000:def 30</a></span>;<br/><a NAME="E6:167_1_1_9_2_1_2"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D15" onmouseover="rs('glib_000/D15')" onmouseout="rh()">GLIB_000:def 15</a></span>;<br/><a NAME="E7:167_1_1_9_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A63</font></span>: 
( ( <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_2"><span class="lab"><font color="Green" title="E98">A58</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E8:167_1_1_9_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A64</font></span>: 
( ( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">thus </span><a NAME="E9:167_1_1_9_2_1_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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_9_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c19">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:167_1_1_9_2_1_2_1"/><span class="lab"><font color="Green" title="E103">A65</font></span>: 
<font color="Maroon" title="c19">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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="E3:167_1_1_9_2_1_2_1"/><span class="lab"><font color="Green" title="E104">A66</font></span>: 
 not <font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_9_2_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_9_2_1_2_1_1"/>
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">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"> contradiction</span><br/>

<a NAME="E2:167_1_1_9_2_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c5">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_2_1_2_1"><span class="lab"><font color="Green" title="E103">A65</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/>
<a NAME="E3:167_1_1_9_2_1_2_1_1"/><span class="kw">then </span>
<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="c5">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:167_1_1_9_2_1_2"><span class="lab"><font color="Green" title="E102">A64</font></span></a>, <a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:167_1_1_9_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</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="E4:167_1_1_9_2_1_2_1"/>
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c15">G4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_2_1_2_1"><span class="lab"><font color="Green" title="E103">A65</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<a NAME="E5:167_1_1_9_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
;<br/>
<a NAME="E6:167_1_1_9_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A67</font></span>: 
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1_9_2_1_2_1"><span class="lab"><font color="Green" title="E104">A66</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font><span class="kw"> such that </span><br/><a NAME="E8:167_1_1_9_2_1_2_1"/><span class="lab"><font color="Green" title="E106">A68</font></span>: 
<font color="Maroon" title="c19">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c20">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c15">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font>

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E6:167_1_1_9_2_1_2_1"><span class="lab"><font color="Green" title="E105">A67</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c20">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E6:167_1_1_9_2_1_2_1"><span class="lab"><font color="Green" title="E105">A67</font></span></a>, <a class="txt" href="#E8:167_1_1_9_2_1_2_1"><span class="lab"><font color="Green" title="E106">A68</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c20">x</font>

;<br/>
<a NAME="E10:167_1_1_9_2_1_2_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c15">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font> <a href="tarski.html#R2" title="TARSKI: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;  not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c15">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_2_1_2_1_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_1_9_2_1_2_1_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI: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;  not <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI: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;  not <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_2_1_2_1"><span class="lab"><font color="Green" title="E103">A65</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_2_1_2_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_9_2_1_2_1_3_1"/>
( <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI: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;  not <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:167_1_1_9_2_1_2_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E7:167_1_1_9_2_1_2"><span class="lab"><font color="Green" title="E101">A63</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_2_1_2_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_9_2_1_2_1_3_2"/>
( <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI: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;  not <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:167_1_1_9_2_1_2_1_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E7:167_1_1_9_2_1_2"><span class="lab"><font color="Green" title="E101">A63</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E6:167_1_1_9"/><span class="lab"><font color="Green" title="E98">A69</font></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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><a NAME="E7:167_1_1_9"/><span class="lab"><font color="Green" title="E99">A70</font></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="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_9"><span class="lab"><font color="Green" title="E98">A69</font></span></a>, <a class="ref" href="glib_006.html#T70" onmouseover="rs('glib_006/T70')" onmouseout="rh()">GLIB_006:70</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_1_9_3"/>
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or ( <font color="Maroon" title="c16">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_3_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_9_3_1"/><span class="lab"><font color="Green" title="E100">A71</font></span>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><div class="add"><a NAME="E2:167_1_1_9_3_1"/><span class="lab"><font color="Green" title="E101">A72</font></span>: 
 not <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1_9"><span class="lab"><font color="Green" title="E97">A57</font></span></a>, <a class="txt" href="#E1:167_1_1_9_3_1"><span class="lab"><font color="Green" title="E100">A71</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:167_1_1_9_3_1"/>
<font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_9_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_9_3_1_1"/><span class="lab"><font color="Green" title="E102">A73</font></span>: 
 not <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"> contradiction</span><br/>

<a NAME="E2:167_1_1_9_3_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/>
<a NAME="E3:167_1_1_9_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_3_1"><span class="lab"><font color="Green" title="E101">A72</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:167_1_1_9_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A74</font></span>: 
<font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E102">A73</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">e1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E5:167_1_1_9_3_1_1"/><span class="lab"><font color="Green" title="E104">A75</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E103">A74</font></span></a></span>;<br/>
<a NAME="E6:167_1_1_9_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A76</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span>
 
;<br/>
<a NAME="E7:167_1_1_9_3_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E104">A75</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E8:167_1_1_9_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A77</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c15">G4</font>
 
;<br/>
<span class="lab"><font color="Green" title="E107">A78</font></span>: <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c15">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E104">A75</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E104">A75</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">w</font>

;<br/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c15">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E105">A76</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E104">A75</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/>
<a NAME="E11:167_1_1_9_3_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c15">G4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E106">A77</font></span></a>, <a class="txt" href="#E9:167_1_1_9_3_1_1"><span class="lab"><font color="Green" title="E107">A78</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:167_1_1_9_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_9"><span class="lab"><font color="Green" title="E98">A69</font></span></a>, <a class="txt" href="#E1:167_1_1_9_3_1"><span class="lab"><font color="Green" title="E100">A71</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/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:167_1_1_9_3_1"/><span class="lab"><font color="Green" title="E102">A79</font></span>: 
( <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E0</font> &amp; <font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> )
 <span class="kw">and </span><br/><a NAME="E6:167_1_1_9_3_1"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> holds <br/><font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167"><span class="lab"><font color="Green" title="E83">A2</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">e1</font> = <font color="Maroon" title="c18">e1</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="c19">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</font></span><br/><span class="kw">thus </span><a NAME="E7:167_1_1_9_3_1"/>
<font color="Maroon" title="c19">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_9_3_1"><span class="lab"><font color="Green" title="E100">A71</font></span></a>, <a class="txt" href="#E5:167_1_1_9_3_1"><span class="lab"><font color="Green" title="E102">A79</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_3_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_9_3_2"/><span class="lab"><font color="Green" title="E100">A80</font></span>: 
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><div class="add"><a NAME="E2:167_1_1_9_3_2"/><span class="lab"><font color="Green" title="E101">A81</font></span>: 
 not <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1_9"><span class="lab"><font color="Green" title="E97">A57</font></span></a>, <a class="txt" href="#E1:167_1_1_9_3_2"><span class="lab"><font color="Green" title="E100">A80</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:167_1_1_9_3_2"/>
<font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_1_9_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:167_1_1_9_3_2_1"/><span class="lab"><font color="Green" title="E102">A82</font></span>: 
 not <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"> contradiction</span><br/>

<a NAME="E2:167_1_1_9_3_2_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/>
<a NAME="E3:167_1_1_9_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_3_2"><span class="lab"><font color="Green" title="E101">A81</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:167_1_1_9_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A83</font></span>: 
<font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E102">A82</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">e1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E5:167_1_1_9_3_2_1"/><span class="lab"><font color="Green" title="E104">A84</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E103">A83</font></span></a></span>;<br/>
<a NAME="E6:167_1_1_9_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A85</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span>
 
;<br/>
<a NAME="E7:167_1_1_9_3_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E104">A84</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E8:167_1_1_9_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A86</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c15">G4</font>
 
;<br/>
<span class="lab"><font color="Green" title="E107">A87</font></span>: <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c15">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E104">A84</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:167_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E104">A84</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">u</font>

;<br/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c15">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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> <font color="Maroon" title="c3">V</font> : verum </span> } </span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E105">A85</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E104">A84</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/>
<a NAME="E11:167_1_1_9_3_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font></span>)</span>,<span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c16">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c15">G4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E106">A86</font></span></a>, <a class="txt" href="#E9:167_1_1_9_3_2_1"><span class="lab"><font color="Green" title="E107">A87</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:167_1_1_9_3_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_9"><span class="lab"><font color="Green" title="E98">A69</font></span></a>, <a class="txt" href="#E1:167_1_1_9_3_2"><span class="lab"><font color="Green" title="E100">A80</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/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:167_1_1_9_3_2"/><span class="lab"><font color="Green" title="E102">A88</font></span>: 
( <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E0</font> &amp; <font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> )
 <span class="kw">and </span><br/><a NAME="E6:167_1_1_9_3_2"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> holds <br/><font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167"><span class="lab"><font color="Green" title="E83">A2</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">e1</font> = <font color="Maroon" title="c18">e1</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="c19">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</font></span><br/><span class="kw">thus </span><a NAME="E7:167_1_1_9_3_2"/>
<font color="Maroon" title="c19">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_1_9_3_2"><span class="lab"><font color="Green" title="E100">A80</font></span></a>, <a class="txt" href="#E5:167_1_1_9_3_2"><span class="lab"><font color="Green" title="E102">A88</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1_9_3_3"><span class="kw">suppose </span></a><a NAME="E1:167_1_1_9_3_3"/>
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><div class="add"><a NAME="E2:167_1_1_9_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A89</font></span>: 
(  not <font color="Maroon" title="c16">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c17">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:167_1_1_9_3_3"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/><a NAME="E4:167_1_1_9_3_3"/><span class="kw">then </span>
( <font color="Maroon" title="c16">u</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c17">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_1_9_3_3"><span class="lab"><font color="Green" title="E100">A89</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:167_1_1_9_3_3"/><span class="lab"><font color="Green" title="E101">A90</font></span>: 
<font color="Maroon" title="c18">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_1_9"><span class="lab"><font color="Green" title="E97">A57</font></span></a>, <a class="txt" href="#E7:167_1_1_9"><span class="lab"><font color="Green" title="E99">A70</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">e1</font> = <font color="Maroon" title="c18">e1</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="c19">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</font></span><br/><span class="kw">thus </span><a NAME="E7:167_1_1_9_3_3"/>
<font color="Maroon" title="c19">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">w</font>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:167_1_1_9_3_3"><span class="lab"><font color="Green" title="E101">A90</font></span></a>, <a class="ref" href="glib_006.html#T70" onmouseover="rs('glib_006/T70')" onmouseout="rh()">GLIB_006:70</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">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E29:167_1_1"/>
<font color="Maroon" title="c15">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_2"/><span class="lab"><font color="Green" title="E84">A91</font></span>: 
<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> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  ex <font color="Olive" title="b1">G4</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">V</font> st <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="c4">G2</font></span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c7">G4</font> =  the    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c7">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c4">G2</font></span><br/><a NAME="E2:167_1_2"/><span class="lab"><font color="Green" title="E85">A92</font></span>: 
<font color="Maroon" title="c7">G4</font> is   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_2"><span class="lab"><font color="Green" title="E84">A91</font></span></a>, <a class="ref" href="glib_007.html#T55" onmouseover="rs('glib_007/T55')" onmouseout="rh()">GLIB_007:55</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="167_1_2_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"> (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c7">G4</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="c4">G2</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c7">G4</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">G2</font> &amp; (  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c7">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c7">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</font> ) ) ) )</span></a><div class="add"><span class="kw">thus </span> <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c7">G4</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="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2"><span class="lab"><font color="Green" title="E85">A92</font></span></a>, <a class="ref" href="glib_006.html#D10" onmouseover="rs('glib_006/D10')" onmouseout="rh()">GLIB_006:def 10</a></span>
<br/>.= 
<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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</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"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c7">G4</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">G2</font> &amp; (  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c7">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c7">G4</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">b<sub>5</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">e1</font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c4">G2</font> ) ) ) )</span><br/><a NAME="E2:167_1_2_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c7">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2"><span class="lab"><font color="Green" title="E85">A92</font></span></a>, <a class="ref" href="glib_006.html#D10" onmouseover="rs('glib_006/D10')" onmouseout="rh()">GLIB_006:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:167_1_2_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c7">G4</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">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</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">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c4">G2</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c7">G4</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">u</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c7">G4</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">b<sub>5</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">e1</font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c4">G2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c9">w</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">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="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w</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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E4:167_1_2_1"/><span class="lab"><font color="Green" title="E86">A93</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w</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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font> ) )</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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font> )</span>
<div class="add"><span class="kw">given </span><font color="Maroon" title="c10">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:167_1_2_1_2"/><span class="lab"><font color="Green" title="E87">A94</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">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">  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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_2_1_2_1"/>
( <font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c1">G1</font> or  not <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2_1_2"><span class="lab"><font color="Green" title="E87">A94</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E88">A95</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">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">  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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</font></span><br/><div class="add"><a NAME="E2:167_1_2_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">u</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c9">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><a NAME="E3:167_1_2_1_2_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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_2_1"><span class="lab"><font color="Green" title="E86">A93</font></span></a>, <a class="txt" href="#E1:167_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E88">A95</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><span class="kw">hence </span><a NAME="E4:167_1_2_1_2_1_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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2"><span class="lab"><font color="Green" title="E85">A92</font></span></a>, <a class="ref" href="glib_006.html#T87" onmouseover="rs('glib_006/T87')" onmouseout="rh()">GLIB_006:87</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E88">A96</font></span>: 
 not <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</font></span><br/><div class="add"><a NAME="E2:167_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E89">A97</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#T59" onmouseover="rs('glib_007/T59')" onmouseout="rh()">GLIB_007:59</a></span>;<br/><a NAME="E3:167_1_2_1_2_1_2"/>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2_1_2"><span class="lab"><font color="Green" title="E87">A94</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E4:167_1_2_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a>, <a class="txt" href="#E2:167_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E89">A97</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:167_1_2_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D30" onmouseover="rs('glib_000/D30')" onmouseout="rh()">GLIB_000:def 30</a></span>;<br/><a NAME="E6:167_1_2_1_2_1_2"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D15" onmouseover="rs('glib_000/D15')" onmouseout="rh()">GLIB_000:def 15</a></span>;<br/><a NAME="E7:167_1_2_1_2_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2_1_2"><span class="lab"><font color="Green" title="E87">A94</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E8:167_1_2_1_2_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> ) or ( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E9:167_1_2_1_2_1_2"/><span class="kw">then </span>
(  not <font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> or  not <font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><a NAME="E10:167_1_2_1_2_1_2"/><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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><span class="kw">hence </span><a NAME="E11:167_1_2_1_2_1_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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2"><span class="lab"><font color="Green" title="E85">A92</font></span></a>, <a class="ref" href="glib_006.html#T87" onmouseover="rs('glib_006/T87')" onmouseout="rh()">GLIB_006:87</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">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E6:167_1_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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c7">G4</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">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><a NAME="E7:167_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A98</font></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="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167_1_2"><span class="lab"><font color="Green" title="E85">A92</font></span></a>, <a class="ref" href="glib_006.html#T87" onmouseover="rs('glib_006/T87')" onmouseout="rh()">GLIB_006:87</a></span>;<br/><a NAME="E8:167_1_2_1"/><span class="lab"><font color="Green" title="E88">A99</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/><a NAME="E9:167_1_2_1"/>
 <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> <font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_2"><span class="lab"><font color="Green" title="E84">A91</font></span></a>, <a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>;<br/><a NAME="E10:167_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A100</font></span>: 
<font color="Maroon" title="c3">V</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:167_1_2_1_3"/>
( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or ( <font color="Maroon" title="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c9">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:167_1_2_1_3_1"/><span class="lab"><font color="Green" title="E90">A101</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><div class="add"><a NAME="E2:167_1_2_1_3_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_2_1"><span class="lab"><font color="Green" title="E86">A93</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:167_1_2_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:167_1_2_1"><span class="lab"><font color="Green" title="E88">A99</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:167_1_2_1_3_1"/><span class="lab"><font color="Green" title="E91">A102</font></span>: 
( <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E0</font> &amp; <font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> )
 <span class="kw">and </span><br/><a NAME="E6:167_1_2_1_3_1"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> holds <br/><font color="Maroon" title="c10">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E10:167_1_2_1"><span class="lab"><font color="Green" title="E89">A100</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c11">e1</font> = <font color="Maroon" title="c10">e1</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="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</font></span><br/><span class="kw">thus </span><a NAME="E7:167_1_2_1_3_1"/>
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_2_1_3_1"><span class="lab"><font color="Green" title="E90">A101</font></span></a>, <a class="txt" href="#E5:167_1_2_1_3_1"><span class="lab"><font color="Green" title="E91">A102</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:167_1_2_1_3_2"/><span class="lab"><font color="Green" title="E90">A103</font></span>: 
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><div class="add"><a NAME="E2:167_1_2_1_3_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_2_1"><span class="lab"><font color="Green" title="E86">A93</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:167_1_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:167_1_2_1"><span class="lab"><font color="Green" title="E88">A99</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:167_1_2_1_3_2"/><span class="lab"><font color="Green" title="E91">A104</font></span>: 
( <font color="Maroon" title="c10">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E0</font> &amp; <font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> )
 <span class="kw">and </span><br/><a NAME="E6:167_1_2_1_3_2"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G2</font> holds <br/><font color="Maroon" title="c10">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E10:167_1_2_1"><span class="lab"><font color="Green" title="E89">A100</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c11">e1</font> = <font color="Maroon" title="c10">e1</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="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</font></span><br/><span class="kw">thus </span><a NAME="E7:167_1_2_1_3_2"/>
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167_1_2_1_3_2"><span class="lab"><font color="Green" title="E90">A103</font></span></a>, <a class="txt" href="#E5:167_1_2_1_3_2"><span class="lab"><font color="Green" title="E91">A104</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2_1_3_3"><span class="kw">suppose </span></a><a NAME="E1:167_1_2_1_3_3"/>
( <font color="Maroon" title="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c9">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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">  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive">b<sub>3</sub></font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">e1</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">G2</font></span><br/><div class="add"><a NAME="E2:167_1_2_1_3_3"/><span class="kw">then </span>
(  not <font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:167_1_2_1_3_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">u</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c9">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:167_1_2_1"><span class="lab"><font color="Green" title="E88">A99</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:167_1_2_1_3_3"/><span class="lab"><font color="Green" title="E90">A105</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:167_1_2_1"><span class="lab"><font color="Green" title="E86">A93</font></span></a>, <a class="txt" href="#E7:167_1_2_1"><span class="lab"><font color="Green" title="E87">A98</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c11">e1</font> = <font color="Maroon" title="c10">e1</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="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</font></span><br/><span class="kw">thus </span><a NAME="E6:167_1_2_1_3_3"/>
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:167_1_2_1_3_3"><span class="lab"><font color="Green" title="E90">A105</font></span></a>, <a class="ref" href="glib_006.html#T70" onmouseover="rs('glib_006/T70')" onmouseout="rh()">GLIB_006:70</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">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:167_1_2"/>
<font color="Maroon" title="c7">G4</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c4">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</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">end;</span></div>

</div>
