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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">G1</font> being    <a href="glib_007.html#M2" title="GLIB_007:mode.2">addAdjVertexToAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b4">G2</font> being    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">e</font>, <font color="Olive" title="b7">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b6">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">v2</font>,<font color="Olive" title="b3">G1</font> iff <font color="Olive" title="b6">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">v2</font>,<font color="Olive" title="b4">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="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_007.html#M2" title="GLIB_007:mode.2">addAdjVertexToAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">G2</font> being    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">e</font>, <font color="Olive" title="b6">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Olive" title="b2">G1</font> iff <font color="Olive" title="b5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Olive" title="b3">G2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</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">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_007.html#M2" title="GLIB_007:mode.2">addAdjVertexToAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b2">G2</font> being    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">e</font>, <font color="Olive" title="b5">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Olive" title="b1">G1</font> iff <font color="Olive" title="b4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Olive" title="b2">G2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G1</font> be    <a href="glib_007.html#M2" title="GLIB_007:mode.2">addAdjVertexToAll</a> of <font color="Maroon" title="c1">G</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">G2</font> being    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">e</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c4">G1</font> iff <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Olive" title="b1">G2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G2</font> be    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G</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">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> iff <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> )</span><br/>









<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_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="E1:63_1"/>
( ( <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">G</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="c1">G</font> ) or  not <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">G</font> or <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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_1"/>
( <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">G</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="c1">G</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> iff <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> )</span><br/><div class="add"><a NAME="E2:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">G2</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</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_007.html#T35" target="_self" onmouseover="rs('glib_007/T35')" onmouseout="rh()">Th35</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_1"/>
 for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> iff <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T9" target="_self" onmouseover="rs('glib_007/T9')" onmouseout="rh()">Th9</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="63_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_2"/>
(  not <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">G</font> or <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">G</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> iff <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> )</span><br/><div class="add"><a NAME="E2:63_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c5">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#D2" target="_self" onmouseover="rs('glib_007/D2')" onmouseout="rh()">Def2</a>, <a class="ref" href="glib_007.html#D3" target="_self" onmouseover="rs('glib_007/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E3:63_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T85" onmouseover="rs('glib_000/T85')" onmouseout="rh()">GLIB_000:85</a></span>;<br/><span class="kw">hence </span><a NAME="E4:63_1_2"/>
 for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> iff <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T88" onmouseover="rs('glib_000/T88')" onmouseout="rh()">GLIB_000:88</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>

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