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<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <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">  for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_006.html#M2" title="GLIB_006:mode.2">addVertices</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">V</font>  st <font color="Olive" title="b1">V</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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">G2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> not <font color="Olive" title="b2">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <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">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_006.html#M2" title="GLIB_006:mode.2">addVertices</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c2">V</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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">G2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> not <font color="Olive" title="b1">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_006.html#M2" title="GLIB_006:mode.2">addVertices</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">V</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"> ( <font color="Maroon" title="c2">V</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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">G2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  not <font color="Maroon" title="c3">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:16"/>
<font color="Maroon" title="c2">V</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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">G2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="c3">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> </span><br/>

<span class="kw">then </span><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="E3:16"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c4">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="glib_006.html#T86" onmouseover="rs('glib_006/T86')" onmouseout="rh()">GLIB_006:86</a></span>;<br/>
<a NAME="E5:16"/>
<font color="Maroon" title="c5">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="glib_006.html#T88" onmouseover="rs('glib_006/T88')" onmouseout="rh()">GLIB_006:88</a>, <a class="ref" href="glib_000.html#D49" onmouseover="rs('glib_000/D49')" onmouseout="rh()">GLIB_000:def 49</a></span>;<br/>
<a NAME="E6:16"/><span class="kw">then </span>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16"/>
 not <font color="Maroon" title="c3">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> 
 ; <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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