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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">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><a NAME="E1:159"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:159"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="159_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:159_1_1"/>
( ( <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">v2</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">v1</font> ) or ( <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">v1</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">v2</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_1"><b>suppose </b></a><a NAME="E1:159_1_1_1"/><i><font color="Green">E30</font></i>: 
( <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">v2</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">v1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:159_1_1_1"/>
<font color="Maroon">v1</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="159_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:159_1_1_1_1_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> or <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:159_1_1_1_1_1_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:159_1_1_1_1_1_1"/>
<font color="Maroon">v2</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:159_1_1_1_1_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:159_1_1_1_1_1_2"/>
<font color="Maroon">v2</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:159_1_1_1"/>
<font color="Maroon">v2</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_2"><b>suppose </b></a><a NAME="E1:159_1_1_2"/><i><font color="Green">E41</font></i>: 
( <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">v1</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">v2</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="159_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:159_1_1_2_1_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> or <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:159_1_1_2_1_1_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:159_1_1_2_1_1_1"/>
<font color="Maroon">v1</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:159_1_1_2_1_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:159_1_1_2_1_1_2"/>
<font color="Maroon">v1</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:159_1_1_2"/>
<font color="Maroon">v1</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><b>thus </b><a NAME="E4:159_1_1_2"/>
<font color="Maroon">v2</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:159"/>
( <font color="Maroon">v1</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v2</font> <a href="glib_000.html#K30" target="_self">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 ;<br/>


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