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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:162_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> 
 ;<br/><a NAME="E2:162_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:162_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>take </b><font color="Maroon">e</font> = <font color="Maroon">e</font>;<br/><a NAME="E5:162_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="txt" href="glib_000.html#E14"><i><font color="Green">Lemma33</font></i></a></i>;<br/><b>hence </b><a NAME="E6:162_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">x</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:162"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">x</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:162"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:162"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> 
 
<b>by </b><i><a class="txt" href="glib_000.html#E14"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E5:162"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a>, </i>;<br/>
<a NAME="E6:162"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:162"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K37" target="_self">.inNeighbors()</a> 
 ;<br/>


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