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<span class="kw">let </span><font color="Maroon" title="c2">H</font> be    <a href="glib_009.html#M4" title="GLIB_009:mode.4">removeDParallelEdges</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">H</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">E</font> being    <a href="glib_009.html#M2" title="GLIB_009:mode.2">RepDEdgeSelection</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:75_2_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c2">H</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_009.html#D8" onmouseover="rs('glib_009/D8')" onmouseout="rh()">GLIB_009:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:75_2_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T136" onmouseover="rs('glib_000/T136')" onmouseout="rh()">GLIB_000:136</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:75_2_1"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c5">e0</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:75_2_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
( <font color="Maroon" title="c6">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> )
 <span class="kw">and </span><br/><a NAME="E9:75_2_1"/>
 for <font color="Olive" title="b1">e9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">e9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> holds <br/><font color="Olive" title="b1">e9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_2_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="glib_009.html#D6" onmouseover="rs('glib_009/D6')" onmouseout="rh()">GLIB_009:def 6</a></span>;<br/>
<a NAME="E10:75_2_1"/><span class="lab"><font color="Green" title="E21">A11</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> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T34" onmouseover="rs('glib_000/T34')" onmouseout="rh()">GLIB_000:34</a></span>;<br/>
<a NAME="E11:75_2_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
;<br/>
<a NAME="E12:75_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E8:75_2_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E10:75_2_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E13:75_2_1"/>
( <font color="Maroon" title="c4">v</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  &amp; <font color="Maroon" title="c6">e</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E14:75_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:75_2_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E12:75_2_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="glib_000.html#T73" onmouseover="rs('glib_000/T73')" onmouseout="rh()">GLIB_000:73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:75_2_1"/>
 not <font color="Maroon" title="c2">H</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#T136" onmouseover="rs('glib_000/T136')" onmouseout="rh()">GLIB_000:136</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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