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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="glib_000.html#M2" target="_self">Subgraph</a> of <font color="Maroon">c<sub>1</sub></font>;<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/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:175"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:175_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:175_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T72" target="_self">Th72</a></i>;<br/><a NAME="E4:175_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:175"><i><font color="Green">E72</font></i></a>, <a class="txt" href="glib_000.html#E3:175_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="glib_000.html#T75" target="_self">Th75</a></i>;<br/><b>hence </b><a NAME="E5:175_1"/>
<font color="Maroon">c<sub>5</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:175"/>
<font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a>  <a href="tarski.html#R1">c=</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:175_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:175_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T73" target="_self">Th73</a></i>;<br/><a NAME="E4:175_2"/>
<font color="Maroon">c<sub>6</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>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:175"><i><font color="Green">E72</font></i></a>, <a class="txt" href="glib_000.html#E3:175_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="glib_000.html#T75" target="_self">Th75</a></i>;<br/><b>hence </b><a NAME="E5:175_2"/>
<font color="Maroon">c<sub>5</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:175"/>
<font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K38" target="_self">.outNeighbors()</a>  <a href="tarski.html#R1">c=</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:175_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K39" target="_self">.allNeighbors()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:175_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T74" target="_self">Th74</a></i>;<br/><a NAME="E4:175_3"/>
<font color="Maroon">c<sub>6</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>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:175"><i><font color="Green">E72</font></i></a>, <a class="txt" href="glib_000.html#E3:175_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="glib_000.html#E41"><i><font color="Green">Lemma25</font></i></a></i>;<br/><b>hence </b><a NAME="E5:175_3"/>
<font color="Maroon">c<sub>5</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> 
 <b>by </b><i><a class="ref" href="glib_000.html#T74" target="_self">Th74</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:175"/>
<font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K39" target="_self">.allNeighbors()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39" target="_self">.allNeighbors()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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