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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_002.html#V1" target="_self">connected</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="E1:33"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">v</font> is <a href="glib_000.html#V11">endvertex</a>
 ;<br/>

<b>then consider </b><font color="Maroon">ev</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">ev</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">ev</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D53">GLIB_000:def 53</a></i>;<br/>
<b>set </b><font color="Maroon">VG</font> =  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">VG2</font> =  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><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">G2</font>, <font color="Maroon">v2'</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v1'</font>, <font color="Maroon">v2</font> = <font color="Maroon">v2'</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E43</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/>;<br/><a NAME="E4:33_1"/>
( <font color="Maroon">v1'</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">v2'</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> )
 ;<br/><a NAME="E5:33_1"/><b>then </b>
( <font color="Maroon">v1'</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> &amp; <font color="Maroon">v2'</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="glib_000.html#T50">GLIB_000:50</a></i>;<br/><a NAME="E6:33_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">v1</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> &amp; not <font color="Maroon">v2</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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:33_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">T</font> being   <a href="glib_001.html#NM10">Trail</a> of <font color="Maroon">W</font>;<br/><a NAME="E9:33_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">T</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#T161">GLIB_001:161</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:33_1_1_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="glib_001.html#K3">.first()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="glib_001.html#K4">.last()</a>  )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T144">GLIB_001:144</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/></div><b>end;</b></div><a NAME="E3:33_1_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="E4:33_1_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="E5:33_1_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><a NAME="E11:33_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">T</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E12:33_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_2"/><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><a NAME="E4:33_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">n1</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:33_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>,<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T10">GLIB_001:10</a></i>;<br/><a NAME="E7:33_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2_1"/><i><font color="Green">E69</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> 
 ;<br/><a NAME="E2:33_1_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ev</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_2_1_1_1"/>
( <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 </span><b>by </b><i>, , , <a class="ref" href="glib_000.html#T68">GLIB_000:68</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">T</font> <a href="glib_001.html#K5">.vertexAt</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D8">GLIB_001:def 8</a></i>;<br/><a NAME="E3:33_1_2_1_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, , </i>;<br/><a NAME="E4:33_1_2_1_1_1_1"/><i><font color="Green">E83</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E7:33_1_2_1_1_1_1"/>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T12">GLIB_001:12</a></i>;<br/><a NAME="E8:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:33_1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E10:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E11:33_1_2_1_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:33_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E13:33_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">T</font> <a href="glib_001.html#K5">.vertexAt</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D8">GLIB_001:def 8</a></i>;<br/><a NAME="E3:33_1_2_1_1_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font>
 <b>by </b><i>, , , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:33_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#T11">GLIB_001:11</a></i>;<br/><a NAME="E5:33_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:33_1_2_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:33_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E8:33_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E9:33_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">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">.edgesInOut()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:33_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">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">G1</font> <a href="glib_000.html#K21">.edgesInOut</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><a class="ref" href="glib_000.html#D42">GLIB_000:def 42</a></i>;<br/><a NAME="E11:33_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<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="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><a class="ref" href="glib_000.html#T38">GLIB_000:38</a></i>;<br/><b>hence </b><a NAME="E12:33_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_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><a NAME="E14:33_1"/><b>then </b>
<font color="Maroon">T</font> <a href="glib_001.html#K14">.edges()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">W'</font> = <font color="Maroon">T</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</font> <b>by </b><i>, <a class="ref" href="glib_001.html#T171">GLIB_001:171</a></i>;<br/><a NAME="E16:33_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#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E17:33_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</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="E5:33"/>
<font color="Maroon">G2</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/>


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