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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">G</font> being <a href="glib_000.html#V2">finite</a> 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>  st <font color="Olive">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">G</font> st <br/>( <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; not <font color="Olive">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 or 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> 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>assume </b><a NAME="E1:49_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:49_1_1_1_1_1"/>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E3:49_1_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:49_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T28">GLIB_000:28</a></i>;<br/><a NAME="E5:49_1_1_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E6:49_1_1_1_1_1"/>
 ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; not <font color="Olive">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i><a class="ref" href="glib_000.html#T29">GLIB_000:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> 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>assume </b><a NAME="E1:49_1_1_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><b>consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:49_1_1_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T24">GLIB_000:24</a></i>;<br/><a NAME="E4:49_1_1_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font>;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font>;<br/><b>thus </b><a NAME="E5:49_1_1_1_2_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>consider </b><font color="Maroon">GRV</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">v</font>;<br/><b>assume </b><a NAME="E2:49_1_1_1_2_1_1"/>
<font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><a NAME="E3:49_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">GRV</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="glib_002.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E4:49_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">GRV</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T51">GLIB_000:51</a></i>;<br/><a NAME="E5:49_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">GRV</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>then reconsider </b><font color="Maroon">GRV'</font> = <font color="Maroon">GRV</font> as  <a href="glib_000.html#V4">trivial</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i><a class="ref" href="glib_000.html#T29">GLIB_000:29</a></i>;<br/><a NAME="E7:49_1_1_1_2_1_1"/>
<font color="Maroon">GRV'</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 ;<br/><b>hence </b><a NAME="E8:49_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:49_1_1_1_2_1"/>
( not <font color="Maroon">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3"><b>suppose </b></a><a NAME="E1:49_1_1_1_3"/>
2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> 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>assume </b><a NAME="E1:49_1_1_1_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:49_1_1_1_3_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1_3_1_1_1"/>
( for <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds not <font color="Olive">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> or  ex <font color="Olive">cv</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <font color="Olive">cv</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds not <font color="Olive">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:49_1_1_1_3_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T24">GLIB_000:24</a></i>;<br/><b>take </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font>;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font>;<br/><b>thus </b><a NAME="E4:49_1_1_1_3_1_1_1_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T1" target="_self">Th1</a></i>;<br/><b>thus </b><a NAME="E5:49_1_1_1_3_1_1_1_1"/>
( not <font color="Maroon">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2"/>
 ex <font color="Olive">cv</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <font color="Olive">cv</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">cv</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">cv</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><b>consider </b><font color="Maroon">G2</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">cv</font>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> </span>)</span>
 ;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> 2 <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2"/><b>then </b>
2 <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><b>then consider </b><font color="Maroon">C1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">C2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  &amp; <font color="Maroon">C2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  &amp; <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C2</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2">PENCIL_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon">C1</font> = <font color="Maroon">C1</font>, <font color="Maroon">C2</font> = <font color="Maroon">C2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  <b>by </b><i><a class="ref" href="glib_002.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E14:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T51">GLIB_000:51</a></i>;<br/><a NAME="E15:49_1_1_1_3_1_1_1_2"/>
<font color="Maroon">G</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">G</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">cv</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">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</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">cv</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/><a NAME="E16:49_1_1_1_3_1_1_1_2"/><b>then </b>
<font color="Maroon">G</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">G</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">cv</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">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">cv</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D42">GLIB_000:def 42</a></i>;<br/><a NAME="E17:49_1_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
(  <a href="glib_000.html#K25">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">the_Vertices_of</a> <font color="Maroon">G</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">cv</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">cv</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>;<br/><div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be   <a href="glib_002.html#NM1">Component</a> of <font color="Maroon">G2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_1"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">a</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">C</font><br/> for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">cv</font>,<font color="Olive">a</font>,<font color="Maroon">G</font>
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">C</font>;<br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x'</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_1"/>
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">cv</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="E8:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">x''</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W</font>;<br/><a NAME="E12:49_1_1_1_3_1_1_1_2_1_1"/>
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">x''</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E13:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E14:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
not <font color="Maroon">P</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/><a NAME="E15:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><a NAME="E16:49_1_1_1_3_1_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E17:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <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">P</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E18:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E19:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 2 <a href="glib_000.html#R1">Joins</a> <font color="Maroon">cv</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>;<br/><a NAME="E20:49_1_1_1_3_1_1_1_2_1_1"/><i><font color="Green">E121</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">x'</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>;<br/><a NAME="E21:49_1_1_1_3_1_1_1_2_1_1"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</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">P</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><a NAME="E22:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">P</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">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E23:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><a NAME="E24:49_1_1_1_3_1_1_1_2_1_1"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <font color="Maroon">P</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">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_1_1"/>
<font color="Maroon">cv</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:49_1_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E163</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E164</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">P</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">cv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</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> 2</span>)</span> &amp; 2 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_1_1"/>
1 <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> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E9:49_1_1_1_3_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> 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#T172">GLIB_001:172</a></i>;<br/><a NAME="E27:49_1_1_1_3_1_1_1_2_1_1"/>
<font color="Maroon">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E28:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">P2</font> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3
 <b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/><a NAME="E29:49_1_1_1_3_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">x'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E30:49_1_1_1_3_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">a</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">C</font>, <font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">cv</font>,<font color="Maroon">a</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1"/>
( <font color="Maroon">C</font> is <a href="glib_000.html#V4">trivial</a> or  not <font color="Maroon">C</font> is <a href="glib_000.html#V4">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1"/>
<font color="Maroon">C</font> is <a href="glib_000.html#V4">trivial</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v''</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_1"/><i><font color="Green">E169</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</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="glib_000.html#T25">GLIB_000:25</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v''</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_1"/>
not <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">cv</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="E7:49_1_1_1_3_1_1_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_1"/><i><font color="Green">E171</font></i>: 
<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> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v'</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><b>consider </b><font color="Maroon">G3</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">v</font>;<br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1"/><i><font color="Green">E173</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G3</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, </i>;<br/><div><i><font color="Green">E174</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">z</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E176</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#T16">GLIB_000:16</a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font> )
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T68">GLIB_000:68</a></i>;<br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b>
<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_002.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">z</font>,<font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>
 <b>by </b><i/>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_2"><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">G3</font>, <font color="Maroon">y</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G3</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_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="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">W</font> = <font color="Maroon">G3</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">x</font>;<br/><b>take </b><font color="Maroon">W</font> = <font color="Maroon">G3</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</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">G</font><b> such that </b><br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W</font>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</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="E8:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/><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> &amp; not <font color="Maroon">y</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="E9:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Maroon">y</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/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</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">P</font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</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/><b>set </b><font color="Maroon">e1</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">e2</font> = <font color="Maroon">P</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>;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <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="E9:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E11:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">P</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="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">P</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> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon">n2</font> <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/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><a NAME="E14:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font>
 <b>by </b><i>, </i>;<br/><a NAME="E15:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T1">GLIB_001:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">P</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> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/>
<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> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><a NAME="E17:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">P</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> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon">n2</font> <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="E19:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">n2</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> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E20:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">cv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">cv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</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">P</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E21:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">cv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">cv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><b>hence </b><a NAME="E22:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><b>thus </b><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2"/>
 ex <font color="Olive">P</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">G3</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_1"/>
not <font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2"/>
not <font color="Maroon">C</font> is <a href="glib_000.html#V4">trivial</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">C'</font> = <font color="Maroon">C</font> as  non <a href="glib_000.html#V4">trivial</a> <a href="glib_000.html#NM2">_Graph</a> ;<br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2"/>
<font color="Maroon">C</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T78">GLIB_000:78</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">C'</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">C'</font><b> such that </b><br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; not <font color="Maroon">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">C'R1</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">C'</font>,<font color="Maroon">v1</font>;<br/><b>consider </b><font color="Maroon">C'R2</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">C'</font>,<font color="Maroon">v2</font>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">C'R1</font> is <a href="glib_002.html#V1" target="_self">connected</a> &amp; <font color="Maroon">C'R2</font> is <a href="glib_002.html#V1" target="_self">connected</a> )
 <b>by </b><i>, </i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2"/>
( <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> )
 <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><a NAME="E11:49_1_1_1_3_1_1_1_2_1_2_1_2"/><b>then </b>
( 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">cv</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">cv</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="E12:49_1_1_1_3_1_1_1_2_1_2_1_2"/><b>then </b><i><font color="Green">E199</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2"/><i><font color="Green">E200</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C'R1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">C'</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">v1</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C'R2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">C'</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">v2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_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="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1"/>
(  not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  or ( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  ) or ( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/>
not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a> 
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T74">GLIB_000:74</a></i>;<br/><b>reconsider </b><font color="Maroon">v'</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="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v'</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><b>consider </b><font color="Maroon">G3</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">v</font>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green">E202</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</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="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/>
not <font color="Maroon">cv</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/><b>then reconsider </b><font color="Maroon">cv'</font> = <font color="Maroon">cv</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G3</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green">E204</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G3</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_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="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">G3</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">y</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_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">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_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="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C'R1</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</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="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C'R1</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">C'R1</font><b> such that </b><br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">y</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E209</font></i>: 
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">W</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">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><i><font color="Green">E210</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1_1"/><b>then </b>
not <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">v1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><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">C</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</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">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</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">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><b>then 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">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172">GLIB_001:172</a></i>;<br/><a NAME="E15:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">y</font>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><a NAME="E16:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <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> 
 <b>by </b><i>, , , <a class="ref" href="glib_000.html#T68">GLIB_000:68</a></i>;<br/><a NAME="E17:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><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">G</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="E18:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><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">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</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="E19:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</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">G</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/><a NAME="E20:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<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">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>;<br/><a NAME="E21:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">cv</font>,<font color="Maroon">a</font>,<font color="Maroon">G3</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><a NAME="E22:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a</font>,<font color="Maroon">cv</font>,<font color="Maroon">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E23:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><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">y</font>,<font color="Maroon">cv</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><a NAME="E24:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E25:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E212</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</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">G</font><b> such that </b><br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W</font>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <font color="Maroon">P</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">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</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="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E217</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon">cv</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">m</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E220</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E221</font></i>: 
( <font color="Maroon">P</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> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <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">P</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><b>then reconsider </b><font color="Maroon">nm1</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E223</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E11:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> 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#T172">GLIB_001:172</a></i>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><b>thus </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_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">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">G3</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">G3</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E14:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"/>
not <font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  )
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E224</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T74">GLIB_000:74</a></i>;<br/><b>reconsider </b><font color="Maroon">v'</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="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v'</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><b>consider </b><font color="Maroon">G3</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">v</font>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/><i><font color="Green">E225</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</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="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/>
not <font color="Maroon">cv</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/><b>then reconsider </b><font color="Maroon">cv'</font> = <font color="Maroon">cv</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G3</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/><i><font color="Green">E226</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G3</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_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="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">G3</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">y</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_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="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E229</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C'R2</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</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="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C'R2</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">C'R2</font><b> such that </b><br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">y</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E231</font></i>: 
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">W</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">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><i><font color="Green">E232</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1_1"/><b>then </b>
not <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">v2</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>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><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">C</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</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">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</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">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><b>then 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">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172">GLIB_001:172</a></i>;<br/><a NAME="E15:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">y</font>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><a NAME="E16:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <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> 
 <b>by </b><i>, , , <a class="ref" href="glib_000.html#T68">GLIB_000:68</a></i>;<br/><a NAME="E17:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><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">G</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="E18:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><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">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</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="E19:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</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">G</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/><a NAME="E20:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<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">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>;<br/><a NAME="E21:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">cv</font>,<font color="Maroon">a</font>,<font color="Maroon">G3</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><a NAME="E22:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a</font>,<font color="Maroon">cv</font>,<font color="Maroon">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E23:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><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">y</font>,<font color="Maroon">cv</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><a NAME="E24:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E25:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E234</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</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">G</font><b> such that </b><br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W</font>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <font color="Maroon">P</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">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</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="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green">E238</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E239</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon">cv</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">m</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E241</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E242</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E243</font></i>: 
( <font color="Maroon">P</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> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <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">P</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><b>then reconsider </b><font color="Maroon">nm1</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E244</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E11:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> 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#T172">GLIB_001:172</a></i>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><b>thus </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">G3</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">G3</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E14:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"/>
not <font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cv</font> <a href="glib_000.html#K39">.allNeighbors()</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">cv</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T74">GLIB_000:74</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><b>set </b><font color="Maroon">a</font> = <font color="Maroon">v2</font>;<br/><b>reconsider </b><font color="Maroon">v'</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="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v'</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><b>consider </b><font color="Maroon">G3</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">v</font>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green">E247</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</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="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/>
not <font color="Maroon">cv</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/><b>then reconsider </b><font color="Maroon">cv'</font> = <font color="Maroon">cv</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G3</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green">E248</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G3</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">G3</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">y</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cv</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_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="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C'R1</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
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">v1</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="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<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">C'R1</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">C'R1</font><b> such that </b><br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">y</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">W</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">W</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><i><font color="Green">E254</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1_1"/><b>then </b>
not <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">v1</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>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><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">C</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</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">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</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">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><b>then 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">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172">GLIB_001:172</a></i>;<br/><a NAME="E15:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">y</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><a NAME="E16:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
not <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> 
 <b>by </b><i>, , , <a class="ref" href="glib_000.html#T68">GLIB_000:68</a></i>;<br/><a NAME="E17:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><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">G</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="E18:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><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">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</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="E19:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</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">G</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/><a NAME="E20:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<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">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50">GLIB_000:50</a></i>;<br/><a NAME="E21:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">cv</font>,<font color="Maroon">v2</font>,<font color="Maroon">G3</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><a NAME="E22:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v2</font>,<font color="Maroon">cv</font>,<font color="Maroon">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><a NAME="E23:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><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">y</font>,<font color="Maroon">cv</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><a NAME="E24:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E25:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E256</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</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">G</font><b> such that </b><br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W</font>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E259</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <font color="Maroon">P</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">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</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="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E261</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon">cv</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">m</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="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E263</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E264</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E5:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E265</font></i>: 
( <font color="Maroon">P</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> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cv</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <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">P</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><b>then reconsider </b><font color="Maroon">nm1</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E266</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E11:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><a NAME="E12:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E13:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> 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#T172">GLIB_001:172</a></i>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
<font color="Maroon">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><b>thus </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/>
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G3</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">cv'</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><a NAME="E14:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon">G3</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/><a NAME="E15:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon">G3</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E16:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"/>
not <font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:49_1_1_1_3_1_1_1_2_1_2_1_2"/>
 ex <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> &amp; not <font color="Olive">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:49_1_1_1_3_1_1_1_2_1"/>
 ex <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">C</font> &amp; not <font color="Olive">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">CC1</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G2</font>,<font color="Maroon">C1</font>;<br/><b>consider </b><font color="Maroon">CC2</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G2</font>,<font color="Maroon">C2</font>;<br/><b>consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E22:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E269</font></i>: 
( <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">CC1</font> &amp; not <font color="Maroon">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:49_1"><i><font color="Green">E41</font></i></a></i>;<br/><b>consider </b><font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E24:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E270</font></i>: 
( <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">CC2</font> &amp; not <font color="Maroon">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:49_1"><i><font color="Green">E41</font></i></a></i>;<br/><b>take </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font>;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_3_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_3_1_1_1_2_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_2"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">CC1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">CC2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:49_1_1_1_3_1_1_1_2_2"/>
( <font color="Maroon">C1</font> is  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#K25">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span> &amp; <font color="Maroon">C2</font> is  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#K25">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_2"/><b>then </b>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">CC1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> &amp;  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">CC2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E5:49_1_1_1_3_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#T1" target="_self">Th1</a>, , <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E26:49_1_1_1_3_1_1_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 ;<br/><b>thus </b><a NAME="E27:49_1_1_1_3_1_1_1_2"/>
( not <font color="Maroon">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:49_1_1_1_3_1"/>
 ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; not <font color="Olive">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:49"/><b>then </b><i><font color="Green">E272</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<a NAME="E3:49"/><i><font color="Green">E273</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> 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/>

<a NAME="E4:49"/>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 
;<br/>
<b>hence </b><a NAME="E5:49"/>
 ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; not <font color="Olive">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i/>;<br/>


</div>
