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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:128"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a> </span><br/>

<a NAME="E2:128"/><span class="kw">then </span>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> or  not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> )
 
;<br/>
<a NAME="E3:128"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D20" target="_self" title="GLIB_000:def.20">Def20</a></span>;<br/>
<a NAME="E4:128"/>
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="glib_000.html#D15" target="_self" title="GLIB_000:def.15">Def15</a></span>;<br/>
<a NAME="E5:128"/><span class="kw">then </span>
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D22" target="_self" title="GLIB_000:def.22">Def22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:128"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128"><span class="lab"><font color="Green" title="E50">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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