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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:164_1"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="164_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:164_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  )
 </span><b>by </b><i><a class="txt" href="glib_000.html#E1:164_1"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_1_1"><b>suppose </b></a><a NAME="E1:164_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:164_1_1_1_1"/><i><font color="Green" title="E68">A2</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T72" target="_self" title="GLIB_000:th.72">Th72</a></i>;<br/><b>take </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">e</font>;<br/><b>thus </b><a NAME="E4:164_1_1_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E3:164_1_1_1_1"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="glib_000.html#E20"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_1_2"><b>suppose </b></a><a NAME="E1:164_1_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:164_1_1_1_2"/><i><font color="Green" title="E68">A3</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T73" target="_self" title="GLIB_000:th.73">Th73</a></i>;<br/><b>take </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">e</font>;<br/><b>thus </b><a NAME="E4:164_1_1_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E3:164_1_1_1_2"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="glib_000.html#E20"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:164_1"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:164"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:164"/><i><font color="Green" title="E67">A4</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="164_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:164_2_1"/>
( <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">G</font> or <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font> )
 </span><b>by </b><i><a class="txt" href="glib_000.html#E4:164"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="glib_000.html#E20"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_2_1_1"><b>suppose </b></a><a NAME="E1:164_2_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><div class="add"><a NAME="E2:164_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T72" target="_self" title="GLIB_000:th.72">Th72</a></i>;<br/><b>hence </b><a NAME="E3:164_2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_2_1_2"><b>suppose </b></a><a NAME="E1:164_2_1_2"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><div class="add"><a NAME="E2:164_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T73" target="_self" title="GLIB_000:th.73">Th73</a></i>;<br/><b>hence </b><a NAME="E3:164_2_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:164"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> 
 ;<br/>


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