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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:119_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">G</font> is <a href="glib_000.html#V3" target="_self">loopless</a>
 ;<br/><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:119_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:119_1_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#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">v</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="ref" href="glib_000.html#D20" target="_self">Def20</a>, </i>;<br/><b>hence </b><a NAME="E3:119_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_000.html#D19" target="_self">Def19</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:119_1"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:119"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Olive">v</font>,<font color="Olive">v</font>,<font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:119_2"/><i><font color="Green">E41</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> <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>set </b><font color="Maroon">v</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/><a NAME="E2:119_2"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</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>,<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>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D20" target="_self">Def20</a>, </i>;<br/><b>hence </b><a NAME="E3:119_2"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_000.html#D19" target="_self">Def19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:119"/>
<font color="Maroon">G</font> is <a href="glib_000.html#V3" target="_self">loopless</a>
 <b>by </b><i/>;<br/>


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