<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 or 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></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">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:49_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></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><a class="txt" href="glib_002.html#E1:49_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="glib_002.html#E1:49_1_1_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E3:49_1_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>2</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_1_1"><i><font color="Green">E37</font></i></a>, <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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></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">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:49_1_1_1_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green">Lemma29</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E5:49_1_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_2_1"><i><font color="Green">E37</font></i></a></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">c<sub>7</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>assume </b><a NAME="E2:49_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></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">E39</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E4:49_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="glib_002.html#E1:49_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="glib_002.html#E1:49_1_1_1_2_1"><i><font color="Green">E36</font></i></a>, <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">c<sub>8</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K5">-</a> 1
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></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><a class="txt" href="glib_002.html#E3:49_1_1_1_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="glib_002.html#E30"><i><font color="Green">Lemma29</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:49_1_1_1_2_1"/>
( not <font color="Maroon">c<sub>5</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>2</sub></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">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:49_1_1_1_3_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green">Lemma29</font></i></a></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">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> holds not <font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></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">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> holds not <font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:49_1_1_1_3_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E4:49_1_1_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>thus </b><a NAME="E5:49_1_1_1_3_1_1_1_1"/>
( not <font color="Maroon">c<sub>5</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">c<sub>6</sub></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_1_1_3_1_1_1_1"><i><font color="Green">E37</font></i></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"><b>suppose </b></a><a NAME="E1:49_1_1_1_3_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E2:49_1_1_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self">Def11</a></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">E38</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E14:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:49_1_1_1_3_1"><i><font color="Green">E35</font></i></a>, <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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">E40</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></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<sub>2</sub></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">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">E41</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<sub>9</sub></font> be   <a href="glib_002.html#NM1">Component</a> of <font color="Maroon">c<sub>4</sub></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">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>9</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>2</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>9</sub></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<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E17:49_1_1_1_3_1_1_1_2"><i><font color="Green">E40</font></i></a>, <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">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">c<sub>13</sub></font>;<br/><a NAME="E12:49_1_1_1_3_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E44</font></i></a>, <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">E45</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></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">E46</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E8:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E43</font></i></a>, <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">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></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">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E15:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E47</font></i></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">c<sub>14</sub></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">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">c<sub>14</sub></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">c<sub>2</sub></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">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E45</font></i></a>, <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">E49</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span>;<br/><a NAME="E21:49_1_1_1_3_1_1_1_2_1_1"/><i><font color="Green">E50</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">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E14:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E46</font></i></a>, <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">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></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">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E45</font></i></a>, <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">E52</font></i>: 
( <font color="Maroon">c<sub>14</sub></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">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E45</font></i></a>, <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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E53</font></i>: 
( <font color="Maroon">c<sub>16</sub></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">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">E54</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></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">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_1_1"><i><font color="Green">E53</font></i></a>, <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">E55</font></i>: 
( <font color="Maroon">c<sub>14</sub></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">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E21:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_1_1"><i><font color="Green">E53</font></i></a>, <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">E56</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; 2 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E6:49_1_1_1_3_1_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <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">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <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="txt" href="glib_002.html#E8:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="glib_002.html#E24:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="glib_002.html#E7:49_1_1_1_3_1_1_1_2_1_1_1"><i><font color="Green">E56</font></i></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">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>16</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 3,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E23:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E51</font></i></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">c<sub>16</sub></font> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E30:49_1_1_1_3_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_002.html#E19:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_002.html#E20:49_1_1_1_3_1_1_1_2_1_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>11</sub></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">E42</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:49_1_1_1_3_1_1_1_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E42</font></i></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"><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<sub>9</sub></font> is <a href="glib_000.html#V4">trivial</a> or  not <font color="Maroon">c<sub>9</sub></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<sub>9</sub></font> is <a href="glib_000.html#V4">trivial</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>9</sub></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">E44</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>12</sub></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<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E17:49_1_1_1_3_1_1_1_2"><i><font color="Green">E40</font></i></a>, <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">E45</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="glib_002.html#E28"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><b>thus </b><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></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">E47</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font>;<br/><a NAME="E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1"/><i><font color="Green">E48</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E2:49_1_1_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="glib_002.html#E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self">Def11</a></i>;<br/><div><i><font color="Green">E49</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">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></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">E50</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>2</sub></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">E51</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></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">E52</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E7:49_1_1_1_3_1_1_1_2_1_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="glib_002.html#E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1"><i><font color="Green">E50</font></i></a>, <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">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E17:49_1_1_1_3_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="glib_002.html#E2:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1"><i><font color="Green">E51</font></i></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">E53</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1"><i><font color="Green">E50</font></i></a>, <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">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E8"><i><font color="Green">Lemma8</font></i></a></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">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_002.html#E9"><i><font color="Green">Lemma9</font></i></a></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="txt" href="glib_002.html#E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="glib_002.html#E8:49_1_1_1_3_1_1_1_2_1_2_1_1"><i><font color="Green">E46</font></i></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">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>16</sub></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">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></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">E50</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">c<sub>17</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">c<sub>17</sub></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">c<sub>20</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_1"><i><font color="Green">E50</font></i></a>, <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">E50</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>21</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></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">E51</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">c<sub>21</sub></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">E52</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E51</font></i></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">c<sub>16</sub></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<sub>2</sub></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">c<sub>15</sub></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">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></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">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></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">E53</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></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">c<sub>22</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E6:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E52</font></i></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">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E54</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></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">c<sub>23</sub></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">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <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">c<sub>23</sub></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">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</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="nat_1.html#NM1">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">c<sub>25</sub></font> = <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></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">c<sub>23</sub></font> <a href="real_1.html#K5">-</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">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <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">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">c<sub>2</sub></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">E55</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <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">E56</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <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">E57</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <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">E58</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></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">c<sub>24</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</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="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="glib_002.html#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">E58</font></i></a>, <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">E58</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E55</font></i></a></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">E59</font></i>: 
<font color="Maroon">c<sub>23</sub></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">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E11:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E56</font></i></a>, <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">E60</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></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">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>23</sub></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="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_002.html#E15:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="glib_002.html#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">E60</font></i></a>, <a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <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">E60</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1_2_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E57</font></i></a></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">c<sub>24</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</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">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>23</sub></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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></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">c<sub>22</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E14:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="glib_002.html#E15:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="glib_002.html#E17:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E60</font></i></a>, <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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E6:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E52</font></i></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><a class="txt" href="glib_002.html#E1:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>16</sub></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">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></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">c<sub>24</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:49_1_1_1_3_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green">E52</font></i></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>16</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></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">c<sub>16</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></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><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_002.html#E30"><i><font color="Green">Lemma29</font></i></a></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">c<sub>15</sub></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<sub>9</sub></font> is <a href="glib_000.html#V4">trivial</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></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<sub>9</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></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<sub>9</sub></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">c<sub>4</sub></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">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>12</sub></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">E44</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">c<sub>14</sub></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">E34</font></i></a>, <a class="txt" href="glib_002.html#E14:49_1_1_1_3_1_1_1_2"><i><font color="Green">E39</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>;<br/><a NAME="E9:49_1_1_1_3_1_1_1_2_1_2_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E6:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="glib_002.html#E33"><i><font color="Green">Lemma32</font></i></a></i>;<br/><a NAME="E10:49_1_1_1_3_1_1_1_2_1_2_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E17:49_1_1_1_3_1_1_1_2"><i><font color="Green">E40</font></i></a>, <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">E46</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">E47</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>15</sub></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<sub>12</sub></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">c<sub>13</sub></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<sub>16</sub></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<sub>12</sub></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">c<sub>14</sub></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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39">.allNeighbors()</a>  or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39">.allNeighbors()</a>  ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">E48</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="glib_000.html#T74">GLIB_000:74</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></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">E49</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>20</sub></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<sub>2</sub></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">c<sub>19</sub></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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>20</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green">E49</font></i></a>, <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">E50</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green">Lemma27</font></i></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"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>20</sub></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">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>20</sub></font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">E51</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> or  not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></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">E52</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green">E49</font></i></a>, <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">E53</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="glib_002.html#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">E52</font></i></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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green">E48</font></i></a>, <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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>15</sub></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">E54</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_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">E53</font></i></a>, <a class="txt" href="glib_002.html#E9:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a></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">E55</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>23</sub></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">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E54</font></i></a>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><i><font color="Green">E56</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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></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">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>9</sub></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">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E9:49_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">E56</font></i></a>, <a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></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">E57</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E55</font></i></a>, <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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green">E48</font></i></a>, <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">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E43</font></i></a>, <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">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>19</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>20</sub></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">c<sub>11</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <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">c<sub>11</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>20</sub></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">c<sub>27</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E57</font></i></a>, <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">c<sub>27</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>22</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">E52</font></i>: 
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>24</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></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">E53</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">c<sub>24</sub></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">E54</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E53</font></i></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_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E54</font></i></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_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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E56</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></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">c<sub>26</sub></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">E57</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#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">E55</font></i></a>, <a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></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">E58</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="txt" href="glib_002.html#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">E55</font></i></a>, <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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E59</font></i>: 
( <font color="Maroon">c<sub>28</sub></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">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">E60</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></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">c<sub>28</sub></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">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E59</font></i></a>, <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">E61</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="txt" href="glib_002.html#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">E59</font></i></a>, <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E57</font></i></a>, <a class="txt" href="glib_002.html#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">E60</font></i></a>, <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="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</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">E62</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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E61</font></i></a></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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E61</font></i></a>, <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="txt" href="glib_002.html#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">E51</font></i></a>, <a class="txt" href="glib_002.html#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">E55</font></i></a>, <a class="txt" href="glib_002.html#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"><i><font color="Green">E61</font></i></a>, <a class="txt" href="glib_002.html#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"><i><font color="Green">E62</font></i></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">c<sub>28</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>28</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E58</font></i></a>, <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="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#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">E52</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></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">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></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">c<sub>27</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E54</font></i></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="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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">c<sub>20</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E18"><i><font color="Green">Lemma17</font></i></a></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">c<sub>20</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green">Lemma29</font></i></a></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">c<sub>19</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:49_1_1_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self">Def11</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_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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="glib_000.html#T74">GLIB_000:74</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></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">E49</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>20</sub></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<sub>2</sub></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">c<sub>19</sub></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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>20</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <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">E50</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green">Lemma27</font></i></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"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>20</sub></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">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>20</sub></font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">c<sub>22</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">E51</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> or  not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></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">E52</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <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">E53</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="glib_002.html#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">E52</font></i></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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>16</sub></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">E54</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_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">E53</font></i></a>, <a class="txt" href="glib_002.html#E9:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a></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">E55</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>23</sub></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">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E54</font></i></a>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><i><font color="Green">E56</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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></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="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></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">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>9</sub></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">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E9:49_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">E56</font></i></a>, <a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></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">E57</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E55</font></i></a>, <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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#E2:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <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">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E43</font></i></a>, <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">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>19</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>20</sub></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">c<sub>11</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <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">c<sub>11</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>20</sub></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">c<sub>27</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E57</font></i></a>, <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">c<sub>27</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>22</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">E52</font></i>: 
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>24</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></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">E53</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">c<sub>24</sub></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">E54</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E53</font></i></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_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E54</font></i></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_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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E56</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></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">c<sub>26</sub></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">E57</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#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">E55</font></i></a>, <a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></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">E58</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="txt" href="glib_002.html#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">E55</font></i></a>, <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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E59</font></i>: 
( <font color="Maroon">c<sub>28</sub></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">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">E60</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></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">c<sub>28</sub></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">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E59</font></i></a>, <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">E61</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="txt" href="glib_002.html#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">E59</font></i></a>, <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E57</font></i></a>, <a class="txt" href="glib_002.html#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">E60</font></i></a>, <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="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</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">E62</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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E61</font></i></a></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">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E61</font></i></a>, <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="txt" href="glib_002.html#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">E51</font></i></a>, <a class="txt" href="glib_002.html#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">E55</font></i></a>, <a class="txt" href="glib_002.html#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"><i><font color="Green">E61</font></i></a>, <a class="txt" href="glib_002.html#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"><i><font color="Green">E62</font></i></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">c<sub>28</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>26</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>28</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E58</font></i></a>, <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="txt" href="glib_002.html#E10:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#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">E52</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></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">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></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">c<sub>27</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E54</font></i></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="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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>20</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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">c<sub>20</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E18"><i><font color="Green">Lemma17</font></i></a></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">c<sub>20</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green">Lemma29</font></i></a></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">c<sub>19</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:49_1_1_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self">Def11</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_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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39">.allNeighbors()</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K39">.allNeighbors()</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>17</sub></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">E48</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></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">E49</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></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">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>21</sub></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">E50</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>22</sub></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<sub>2</sub></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">c<sub>21</sub></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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>22</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green">E50</font></i></a>, <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">E51</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green">Lemma27</font></i></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"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>22</sub></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">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>22</sub></font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">c<sub>24</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>22</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></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">E52</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> or  not <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></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">E53</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></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">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E9:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green">E50</font></i></a>, <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">E54</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="glib_002.html#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">E53</font></i></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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E44</font></i></a>, <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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>15</sub></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">E55</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_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">E54</font></i></a>, <a class="txt" href="glib_002.html#E9:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a></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">E56</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E55</font></i></a>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><i><font color="Green">E57</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">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></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">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></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="txt" href="glib_002.html#E13:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E47</font></i></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">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>9</sub></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">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E9:49_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">E57</font></i></a>, <a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>22</sub></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">E58</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E56</font></i></a>, <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">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#E6:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green">E48</font></i></a>, <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">c<sub>17</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green">E49</font></i></a>, <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">c<sub>17</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>21</sub></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">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>21</sub></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">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>22</sub></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">c<sub>17</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green">E48</font></i></a>, <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">c<sub>17</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>22</sub></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">c<sub>29</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>17</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E58</font></i></a>, <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">c<sub>29</sub></font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>24</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>22</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></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">E53</font></i>: 
not <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>26</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></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">E54</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">c<sub>26</sub></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">E55</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E54</font></i></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_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>27</sub></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">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E55</font></i></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_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">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E57</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></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">c<sub>28</sub></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">E58</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="txt" href="glib_002.html#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">E57</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></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">E59</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E57</font></i></a>, <a class="txt" href="glib_002.html#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">E56</font></i></a>, <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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">E60</font></i>: 
( <font color="Maroon">c<sub>30</sub></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">c<sub>27</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">E61</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>30</sub></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">c<sub>30</sub></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">c<sub>27</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E60</font></i></a>, <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">E62</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#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">E57</font></i></a>, <a class="txt" href="glib_002.html#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">E60</font></i></a>, <a class="ref" href="glib_001.html#T48">GLIB_001:48</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E58</font></i></a>, <a class="txt" href="glib_002.html#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">E61</font></i></a>, <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="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="real_1.html#K5">-</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">E63</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">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="real_1.html#K5">-</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">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E62</font></i></a></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">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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"><i><font color="Green">E62</font></i></a>, <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="txt" href="glib_002.html#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">E52</font></i></a>, <a class="txt" href="glib_002.html#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">E56</font></i></a>, <a class="txt" href="glib_002.html#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"><i><font color="Green">E62</font></i></a>, <a class="txt" href="glib_002.html#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"><i><font color="Green">E63</font></i></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">c<sub>30</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>30</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E59</font></i></a>, <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="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="glib_002.html#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">E53</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>22</sub></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">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></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">c<sub>29</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#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">E55</font></i></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="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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>22</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></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">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>22</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></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">c<sub>22</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E18"><i><font color="Green">Lemma17</font></i></a></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">c<sub>22</sub></font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green">Lemma29</font></i></a></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">c<sub>21</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:49_1_1_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self">Def11</a></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">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>9</sub></font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E22:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>11</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E18:49_1_1_1_3_1_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E24:49_1_1_1_3_1_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>12</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E18:49_1_1_1_3_1_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></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">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E2:49_1_1_1_3_1_1_1_2_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E22:49_1_1_1_3_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_002.html#E24:49_1_1_1_3_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="glib_002.html#E29"><i><font color="Green">Lemma28</font></i></a></i>;<br/><a NAME="E3:49_1_1_1_3_1_1_1_2_2"/>
( <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E25"><i><font color="Green">Lemma24</font></i></a></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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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="txt" href="glib_002.html#E12:49_1_1_1_3_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="glib_002.html#E2:49_1_1_1_3_1_1_1_2_2"><i><font color="Green">E44</font></i></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">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>thus </b><a NAME="E27:49_1_1_1_3_1_1_1_2"/>
( not <font color="Maroon">c<sub>13</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">c<sub>14</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E22:49_1_1_1_3_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_002.html#E24:49_1_1_1_3_1_1_1_2"><i><font color="Green">E43</font></i></a></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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></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">c<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:49"/><b>then </b><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<a NAME="E3:49"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="glib_002.html#E2:49"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a> 
 
;<br/>
<b>hence </b><a NAME="E5:49"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Olive">b<sub>2</sub></font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E3:49"><i><font color="Green">E35</font></i></a></i>;<br/>


</div>
