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<b>let </b><font color="Maroon">G1</font> be  <a href="glib_000.html#V2" target="_self">finite</a> 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">VG1</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">VG2</font> =  <a href="glib_000.html#K25" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font>;<br/>
<b>set </b><font color="Maroon">EG1</font> =  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">EG2</font> =  <a href="glib_000.html#K26" target="_self">the_Edges_of</a> <font color="Maroon">G2</font>;<br/>
<b>set </b><font color="Maroon">EV</font> = <font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> ;<br/>
<a NAME="E1:145"/><i><font color="Green">E28</font></i>: 
(  <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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E2:145"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:145"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D24" target="_self">Def24</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:145"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K25" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="glib_000.html#K25" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<a NAME="E5:145"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><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> <a href="subset_1.html#K4">\/</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K16" target="_self">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="glib_000.html#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E6:145"/>
 <a href="card_1.html#K4">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> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K16" target="_self">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="glib_000.html#D25" target="_self">Def25</a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<b>hence </b><a NAME="E7:145"/>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K16" target="_self">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K16" target="_self">.order()</a> 
 ;<br/>

<a NAME="E8:145"/><i><font color="Green">E30</font></i>: 
<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E9:145"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="glib_000.html#K26" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> 
 
<b>by </b><i><a class="ref" href="glib_000.html#D24" target="_self">Def24</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E10:145"/>
 <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" target="_self">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D24" target="_self">Def24</a>, <a class="txt" href="glib_000.html#E30"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E11:145"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K18" target="_self">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/>
<b>hence </b><a NAME="E12:145"/>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K18" target="_self">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K18" target="_self">.size()</a> 
 ;<br/>


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