<?xml version="1.0"?>
<div class="add">

<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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:157_1"/><i><font color="Green">E28</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> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="157_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:157_1_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">v1</font> or <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> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_1_1_1_1"><b>suppose </b></a><a NAME="E1:157_1_1_1_1"/><i><font color="Green">E29</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>
 ;<br/><div class="add"><b>set </b><font color="Maroon">v2</font> = <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>;<br/><b>reconsider </b><font color="Maroon">v2</font> = <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> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font>;<br/><b>thus </b><a NAME="E3:157_1_1_1_1"/>
<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>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_1_1_1_2"><b>suppose </b></a><a NAME="E1:157_1_1_1_2"/><i><font color="Green">E30</font></i>: 
<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>set </b><font color="Maroon">v2</font> = <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>;<br/><b>reconsider </b><font color="Maroon">v2</font> = <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> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font>;<br/><b>thus </b><a NAME="E3:157_1_1_1_2"/>
<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>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:157_1"/>
 ex <font color="Olive">v2</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v1</font>,<font color="Olive">v2</font>,<font color="Maroon">G</font>
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:157"/><i><font color="Green">E41</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/>

<b>thus </b><a NAME="E3:157"/>
<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>, </i>;<br/>


</div>
