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<b>let </b><font color="Maroon">G1</font> be  non <a href="glib_000.html#V4" target="_self">trivial</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G1</font>,<font color="Maroon">v</font>;<br/>





<b>set </b><font color="Maroon">VG</font> =  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:144_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:144_1"/><b>then </b>
 <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:144_1"/><b>then </b>
 <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E4:144_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T50">CARD_1:50</a></i>;<br/><b>hence </b><a NAME="E5:144_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> ;<br/>
<a NAME="E3:144"/>
<font color="Maroon">G2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G1</font>,<font color="Maroon">V</font>
 
;<br/>
<b>hence </b><a NAME="E4:144"/>
(  <a href="glib_000.html#K25" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K26" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>


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