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<b>let </b><font color="Maroon">G1</font> be  non <a href="glib_000.html#V4">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>assume </b><a NAME="E1:63"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">G2</font> is <a href="glib_002.html#V1" target="_self">connected</a> &amp;  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a>  &amp; not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G1</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:63"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a>  &amp; not <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G1</font> )
 ;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font>;<br/><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:63_1"/><b>then </b>
not <font color="Maroon">x</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="E3:63_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">v3</font> = <font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font>;<br/>
<a NAME="E5:63"/>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E4:63"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_000.html#T70">GLIB_000:70</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v3'</font> = <font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="txt" href="glib_002.html#E7:63"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E7:63"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font>,<font color="Maroon">G1</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E4:63"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_000.html#T70">GLIB_000:70</a></i>;<br/>
<a NAME="E8:63"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font>,<font color="Maroon">v</font>,<font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font>, <font color="Maroon">v2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_1_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_1"><b>suppose </b></a><a NAME="E1:63_2_1_1_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">v1'</font> = <font color="Maroon">v1</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="txt" href="glib_002.html#E7:63"><i><font color="Green">E44</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_1_1_1_1_1"/>
( <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:63_2_1_1_1_1_1_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">v2'</font> = <font color="Maroon">v2</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="txt" href="glib_002.html#E7:63"><i><font color="Green">E44</font></i></a></i>;<br/><b>consider </b><font color="Maroon">W'</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E4:63_2_1_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">W'</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1'</font>,<font color="Maroon">v2'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:63"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W'</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E6:63_2_1_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E7:63_2_1_1_1_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:63_2_1_1_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">W'</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E3:63_2_1_1_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">W'</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1'</font>,<font color="Maroon">v3'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:63"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W'</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E5:63_2_1_1_1_1_1_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E6:63_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E7:63_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/><b>hence </b><a NAME="E8:63_2_1_1_1_1_1_2"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:63_2_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_2"><b>suppose </b></a><a NAME="E1:63_2_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_1_1_2_1_1"/>
( <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:63_2_1_1_2_1_1_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">v2'</font> = <font color="Maroon">v2</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="txt" href="glib_002.html#E7:63"><i><font color="Green">E44</font></i></a></i>;<br/><b>set </b><font color="Maroon">W1</font> = <font color="Maroon">G1</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span>;<br/><a NAME="E3:63_2_1_1_2_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">G1</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><b>consider </b><font color="Maroon">W2'</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E5:63_2_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">W2'</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v3'</font>,<font color="Maroon">v2'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:63"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W2'</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E7:63_2_1_1_2_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>take </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E8:63_2_1_1_2_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:63_2_1_1_2_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">W</font> = <font color="Maroon">G1</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E2:63_2_1_1_2_1_1_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, , <a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:63_2_1_1_2"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:63_2"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:63"/>
<font color="Maroon">G1</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/>


</div>
