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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:164_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39" target="_self">.allNeighbors()</a> 
 ;<br/><a NAME="E2:164_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a> </span>)</span>
 ;<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">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a>  )
 </span><b>by </b><i><a class="txt" href="glib_000.html#E2:164_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D2">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">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:164_1_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T72" target="_self">Th72</a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E4:164_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E3:164_1_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="glib_000.html#E22"><i><font color="Green">Lemma10</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">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:164_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T73" target="_self">Th73</a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E4:164_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E3:164_1_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="glib_000.html#E22"><i><font color="Green">Lemma10</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="E4:164_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:164"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:164"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="glib_000.html#E4:164"><i><font color="Green">E69</font></i></a>, <a class="txt" href="glib_000.html#E22"><i><font color="Green">Lemma10</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">c<sub>4</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:164_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T72" target="_self">Th72</a></i>;<br/><b>hence </b><a NAME="E3:164_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">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">c<sub>4</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:164_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T73" target="_self">Th73</a></i>;<br/><b>hence </b><a NAME="E3:164_2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">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">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39" target="_self">.allNeighbors()</a> 
 ;<br/>


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