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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">G</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> non <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v2</font> &amp; not <font color="Olive" title="b2">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b3">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] )</span><br/><b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E33">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> non <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> implies  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> ) )</span><br/><b>assume </b><a NAME="E1:48_1_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><a NAME="E2:48_1_1"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green" title="E29">Lm18</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_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="E1:48_1_1_1_1"/>
( for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font>  holds not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> or  ex <font color="Olive" title="b1">cv</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <font color="Olive" title="b1">cv</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font>  holds not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E3:48_1_1_1_1_1"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T24" title="GLIB_000:th.24">GLIB_000:24</a></i>;<br/><b>take </b><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c3">v1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v2</font> &amp; not <font color="Maroon" title="c5">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b1">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>take </b><font color="Maroon" title="c6">v2</font> = <font color="Maroon" title="c4">v2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font> &amp; not <font color="Maroon" title="c5">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c6">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>thus </b><a NAME="E4:48_1_1_1_1_1"/>
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_1"><i><font color="Green" title="E37">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c5">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c6">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>thus </b><a NAME="E5:48_1_1_1_1_1"/>
( not <font color="Maroon" title="c5">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c6">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="#E1:48_1_1_1_1_1"><i><font color="Green" title="E36">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">cv</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <font color="Olive" title="b1">cv</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">cv</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E3:48_1_1_1_1_2"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c3">cv</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 ;<br/><b>consider </b><font color="Maroon" title="c4">G2</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">cv</font>;<br/><a NAME="E5:48_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <b>by </b><i><a class="txt" href="#E2:48_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></i>;<br/><a NAME="E6:48_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:48_1_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/><a NAME="E8:48_1_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/><a NAME="E9:48_1_1_1_1_2"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">C1</font>, <font color="Maroon" title="c6">C2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:48_1_1_1_1_2"/><i><font color="Green" title="E37">A7</font></i>: 
( <font color="Maroon" title="c5">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Maroon" title="c6">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Maroon" title="c5">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">C2</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">C1</font> = <font color="Maroon" title="c5">C1</font>, <font color="Maroon" title="c8">C2</font> = <font color="Maroon" title="c6">C2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  <b>by </b><i><a class="txt" href="#E11:48_1_1_1_1_2"><i><font color="Green" title="E37">A7</font></i></a></i>;<br/><a NAME="E13:48_1_1_1_1_2"/><i><font color="Green" title="E38">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="#E1:48_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="glib_000.html#T51" title="GLIB_000:th.51">GLIB_000:51</a></i>;<br/><a NAME="E14:48_1_1_1_1_2"/>
<font color="Maroon" title="c2">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T38" title="GLIB_000:th.38">GLIB_000:38</a></i>;<br/><a NAME="E15:48_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D42" title="GLIB_000:def.42">GLIB_000:def 42</a></i>;<br/><a NAME="E16:48_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E39">A9</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><div><i><font color="Green" title="E40">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">C</font> be   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c4">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_1_1_2_1_1"/><i><font color="Green" title="E41">A11</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">C</font><br/> for <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">cv</font>,<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c10">x</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">C</font>;<br/><a NAME="E3:48_1_1_1_1_2_1_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c10">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:48_1_1_1_1_2_1_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">x''</font> = <font color="Maroon" title="c11">x'</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E7:48_1_1_1_1_2_1_1"/>
not <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E16:48_1_1_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E10:48_1_1_1_1_2_1_1"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c13">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c12">x''</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">P</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">W</font>;<br/><a NAME="E12:48_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c14">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c12">x''</font>
 <b>by </b><i><a class="txt" href="#E10:48_1_1_1_1_2_1_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><a NAME="E13:48_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E44">A14</font></i>: 
( <font color="Maroon" title="c14">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <font color="Maroon" title="c14">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E14:48_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E45">A15</font></i>: 
not <font color="Maroon" title="c14">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="#E8:48_1_1_1_1_2_1_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/><a NAME="E15:48_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E46">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/><a NAME="E16:48_1_1_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E17:48_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font>
 <b>by </b><i><a class="txt" href="#E15:48_1_1_1_1_2_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E18:48_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E19:48_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></i>;<br/><a NAME="E20:48_1_1_1_1_2_1_1"/><i><font color="Green" title="E48">A18</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">x'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green" title="E27">Lm16</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c15">P2</font> = <font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span>;<br/><a NAME="E21:48_1_1_1_1_2_1_1"/><i><font color="Green" title="E49">A19</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font>
 <b>by </b><i><a class="txt" href="#E14:48_1_1_1_1_2_1_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/><a NAME="E22:48_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E23:48_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>;<br/><a NAME="E24:48_1_1_1_1_2_1_1"/><i><font color="Green" title="E51">A21</font></i>: 
( <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c16">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E52">A22</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E53">A23</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:48_1_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">P</font> )
 <b>by </b><i><a class="txt" href="#E21:48_1_1_1_1_2_1_1"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="glib_001.html#T48" title="GLIB_001:th.48">GLIB_001:48</a></i>;<br/><a NAME="E7:48_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E54">A24</font></i>: 
( <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_1_1"><i><font color="Green" title="E53">A23</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E9:48_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:48_1_1_1_1_2_1_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="#E24:48_1_1_1_1_2_1_1"><i><font color="Green" title="E51">A21</font></i></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A24</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c16">P2</font> = <font color="Maroon" title="c14">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">P</font></span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><a NAME="E27:48_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c16">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="#E23:48_1_1_1_1_2_1_1"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><a NAME="E28:48_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">P2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
 <b>by </b><i><a class="ref" href="glib_001.html#T24" title="GLIB_001:th.24">GLIB_001:24</a></i>;<br/><a NAME="E29:48_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">x'</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E30:48_1_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:48_1_1_1_1_2_1_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="#E19:48_1_1_1_1_2_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="#E20:48_1_1_1_1_2_1_1"><i><font color="Green" title="E48">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c10">a</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">C</font>, <font color="Maroon" title="c11">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1"/><i><font color="Green" title="E41">A25</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c2">G</font>
 ;<br/><a NAME="E4:48_1_1_1_1_2_1"/><i><font color="Green" title="E42">A26</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1"/>
( <font color="Maroon" title="c9">C</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> or  not <font color="Maroon" title="c9">C</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c9">C</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">v''</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">C</font><b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_1"/><i><font color="Green" title="E43">A27</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">v''</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T25" title="GLIB_000:th.25">GLIB_000:25</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">v'</font> = <font color="Maroon" title="c12">v''</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_1"/>
not <font color="Maroon" title="c13">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E16:48_1_1_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E44">A28</font></i>: 
<font color="Maroon" title="c13">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_1"/><i><font color="Green" title="E45">A29</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c13">v'</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1"><i><font color="Green" title="E43">A27</font></i></a>, <a class="txt" href="glib_002.html#E28"><i><font color="Green" title="E27">Lm16</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">v</font> = <font color="Maroon" title="c13">v'</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Maroon" title="c15">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>thus </b><a NAME="E10:48_1_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c15">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1"/><i><font color="Green" title="E46">A30</font></i>: 
<font color="Maroon" title="c15">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c16">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c15">v</font>;<br/><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1"/><i><font color="Green" title="E47">A31</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">G3</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <b>by </b><i><a class="txt" href="#E2:48_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1"><i><font color="Green" title="E46">A30</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></i>;<br/><div><i><font color="Green" title="E48">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font>, <font color="Maroon" title="c18">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c18">z</font>,<font color="Maroon" title="c2">G</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> )</span><br/><b>assume </b><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_1"/><i><font color="Green" title="E49">A33</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c18">z</font>,<font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> )</span><br/><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E50">A34</font></i>: 
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#T16" title="GLIB_000:th.16">GLIB_000:16</a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_1_1"/><i><font color="Green" title="E51">A35</font></i>: 
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c18">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_1"><i><font color="Green" title="E44">A28</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_1"><i><font color="Green" title="E49">A33</font></i></a>, <a class="ref" href="glib_000.html#T68" title="GLIB_000:th.68">GLIB_000:68</a></i>;<br/><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">G2</font>
 <b>by </b><i><a class="txt" href="#E16:48_1_1_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="#E2:48_1_1_1_1_2_1_2_1_1_1_1"><i><font color="Green" title="E50">A34</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c18">z</font>,<font color="Maroon" title="c4">G2</font>
 <b>by </b><i><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_1"><i><font color="Green" title="E49">A33</font></i></a>, <a class="ref" href="glib_000.html#T76" title="GLIB_000:th.76">GLIB_000:76</a></i>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c13">v'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E8"><i><font color="Green" title="E8">Lm1</font></i></a>, <a class="txt" href="glib_002.html#E9"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_1"><i><font color="Green" title="E45">A29</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_1_1"><i><font color="Green" title="E51">A35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_1"/>
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c16">G3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c16">G3</font> st <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_1"/><i><font color="Green" title="E49">A36</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c16">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c19">W</font> = <font color="Maroon" title="c16">G3</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c17">x</font>;<br/><b>take </b><font color="Maroon" title="c20">W</font> = <font color="Maroon" title="c16">G3</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c17">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><br/><b>thus </b><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c20">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_1"><i><font color="Green" title="E49">A36</font></i></a>, <a class="ref" href="glib_001.html#T14" title="GLIB_001:th.14">GLIB_001:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><i><font color="Green" title="E49">A37</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c16">G3</font> st <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c19">x'</font> = <font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c20">y'</font> = <font color="Maroon" title="c18">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><i><font color="Green" title="E50">A38</font></i>: 
<font color="Maroon" title="c21">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c19">x'</font>,<font color="Maroon" title="c20">y'</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">P</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c21">W</font>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><i><font color="Green" title="E51">A39</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c19">x'</font>,<font color="Maroon" title="c20">y'</font>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E50">A38</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c16">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><b>then </b>
( not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">v</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E52">A40</font></i>: 
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c15">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c23">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E53">A41</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font> &amp; <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="#E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A40</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">n2</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>set </b><font color="Maroon" title="c25">e1</font> = <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c26">e2</font> = <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E54">A42</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n2</font>,<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E55">A43</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="#E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A40</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E56">A44</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E57">A45</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c24">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A40</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E57">A45</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E13:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E57">A46</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1"><i><font color="Green" title="E48">A32</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E54">A42</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><a NAME="E14:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E58">A47</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="#E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E55">A43</font></i></a>, <a class="ref" href="glib_001.html#T1" title="GLIB_001:th.1">GLIB_001:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/><i><font color="Green" title="E59">A48</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A40</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="txt" href="#E14:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E58">A47</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"><i><font color="Green" title="E59">A48</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E16:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E59">A49</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1"><i><font color="Green" title="E48">A32</font></i></a>, <a class="txt" href="#E11:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E56">A44</font></i></a></i>;<br/><a NAME="E17:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c24">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E18:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c24">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E19:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">P</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E57">A46</font></i></a>, <a class="txt" href="#E14:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E58">A47</font></i></a>, <a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E59">A49</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E20:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> )
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><b>hence </b><a NAME="E21:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E49">A37</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c23">P</font> = <font color="Maroon" title="c22">P</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c16">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon" title="c24">P</font> = <font color="Maroon" title="c23">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c24">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><br/><b>thus </b><a NAME="E13:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c16">G3</font> st <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">G3</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E7:48_1_1_1_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1"><i><font color="Green" title="E47">A31</font></i></a>, <a class="txt" href="glib_002.html#E30"><i><font color="Green" title="E29">Lm18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E12:48_1_1_1_1_2_1_2_1_1"/>
not <font color="Maroon" title="c15">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2"/>
not <font color="Maroon" title="c9">C</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c12">C'</font> = <font color="Maroon" title="c9">C</font> as  non <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> ;<br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c9">C</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T78" title="GLIB_000:th.78">GLIB_000:78</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">C</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">v1</font>, <font color="Maroon" title="c14">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C'</font><b> such that </b><br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2"/><i><font color="Green" title="E43">A50</font></i>: 
( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">v2</font> &amp; not <font color="Maroon" title="c13">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c14">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="#E1:48_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="#E13:48_1_1_1_1_2"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">C'R1</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c12">C'</font>,<font color="Maroon" title="c13">v1</font>;<br/><b>consider </b><font color="Maroon" title="c16">C'R2</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c12">C'</font>,<font color="Maroon" title="c14">v2</font>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2"/><i><font color="Green" title="E44">A51</font></i>: 
( <font color="Maroon" title="c15">C'R1</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> &amp; <font color="Maroon" title="c16">C'R2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> )
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E43">A50</font></i></a>, <a class="txt" href="glib_002.html#E32"><i><font color="Green" title="E31">Lm20</font></i></a></i>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_2"/>
( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> &amp; <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_2"/><b>then </b>
( not <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="#E16:48_1_1_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E45">A52</font></i>: 
( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font> &amp; <font color="Maroon" title="c14">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2"/><i><font color="Green" title="E46">A53</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">C'R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c12">C'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c16">C'R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c12">C'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1"/>
(  not <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  or ( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  &amp; not <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  ) or ( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  &amp; <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1"/>
not <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E47">A54</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="glib_000.html#T74" title="GLIB_000:th.74">GLIB_000:74</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">v'</font> = <font color="Maroon" title="c13">v1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c17">v'</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon" title="c19">v</font> = <font color="Maroon" title="c18">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Maroon" title="c19">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>thus </b><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c19">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a></span><br/><b>consider </b><font color="Maroon" title="c20">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c19">v</font>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green" title="E48">A55</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c20">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_1"/>
not <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">cv'</font> = <font color="Maroon" title="c3">cv</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c20">G3</font> <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green" title="E48">A55</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green" title="E49">A56</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c17">v'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green" title="E27">Lm16</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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" title="c22">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c20">G3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1"/>
( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> or <font color="Maroon" title="c22">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">G3</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c22">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14" title="GLIB_001:th.14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"/><i><font color="Green" title="E50">A57</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> or  not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E51">A58</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green" title="E48">A55</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:48_1_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" title="E52">A59</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">C'R1</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E51">A58</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:48_1_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green" title="E47">A54</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">C'R1</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c15">C'R1</font><b> such that </b><br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E53">A60</font></i>: 
<font color="Maroon" title="c23">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E44">A51</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A59</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E54">A61</font></i>: 
( <font color="Maroon" title="c23">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Maroon" title="c23">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> )
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E53">A60</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><i><font color="Green" title="E55">A62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_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" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E3:48_1_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" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c24">W</font> = <font color="Maroon" title="c23">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">C</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">W</font> = <font color="Maroon" title="c24">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">W</font> = <font color="Maroon" title="c25">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">W</font> = <font color="Maroon" title="c26">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E56">A63</font></i>: 
<font color="Maroon" title="c27">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A61</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E16:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="txt" href="#E2:48_1_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green" title="E47">A54</font></i></a>, <a class="ref" href="glib_000.html#T68" title="GLIB_000:th.68">GLIB_000:68</a></i>;<br/><a NAME="E17:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E18:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D42" title="GLIB_000:def.42">GLIB_000:def 42</a></i>;<br/><a NAME="E19:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T38" title="GLIB_000:th.38">GLIB_000:38</a></i>;<br/><a NAME="E20:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c20">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E21:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c20">G3</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="glib_000.html#T76" title="GLIB_000:th.76">GLIB_000:76</a></i>;<br/><a NAME="E22:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c20">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><a NAME="E23:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c27">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c3">cv</font>
 <b>by </b><i><a class="txt" href="#E15:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E56">A63</font></i></a>, <a class="ref" href="glib_001.html#T67" title="GLIB_001:th.67">GLIB_001:67</a></i>;<br/><a NAME="E24:48_1_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" title="c27">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24" title="GLIB_001:th.24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E25:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E51">A64</font></i>: 
not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c23">y'</font> = <font color="Maroon" title="c22">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E52">A65</font></i>: 
<font color="Maroon" title="c24">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c25">P</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E53">A66</font></i>: 
<font color="Maroon" title="c25">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A65</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E54">A67</font></i>: 
( <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y'</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c26">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E55">A68</font></i>: 
( <font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font> &amp; <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_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" title="E56">A69</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font>
 <b>by </b><i><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A67</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E55">A68</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>set </b><font color="Maroon" title="c27">P2</font> = <font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E57">A70</font></i>: 
<font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A67</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E55">A68</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c3">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c28">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E58">A71</font></i>: 
( <font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_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" title="c28">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E58">A71</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:48_1_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" title="E59">A72</font></i>: 
( <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">P</font> )
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E55">A68</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E58">A71</font></i></a>, <a class="ref" href="glib_001.html#T48" title="GLIB_001:th.48">GLIB_001:48</a></i>;<br/><a NAME="E7:48_1_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#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font>
 <b>by </b><i><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E56">A69</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E8:48_1_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></i>;<br/><a NAME="E9:48_1_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" title="E60">A73</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E10:48_1_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" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E59">A72</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:48_1_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="#E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"><i><font color="Green" title="E50">A57</font></i></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A67</font></i></a>, <a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E59">A72</font></i></a>, <a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E60">A73</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c28">P2</font> = <font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c28">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E57">A70</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E10:48_1_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="#E10:48_1_1_1_1_2_1_2_1_2_1_1_1"><i><font color="Green" title="E49">A56</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E51">A64</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c26">P</font> = <font color="Maroon" title="c25">P</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon" title="c27">P</font> = <font color="Maroon" title="c26">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><b>thus </b><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c27">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E53">A66</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">G3</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E18"><i><font color="Green" title="E17">Lm6</font></i></a></i>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">G3</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green" title="E29">Lm18</font></i></a></i>;<br/><b>hence </b><a NAME="E14:48_1_1_1_1_2_1_2_1_2_1_1_1"/>
not <font color="Maroon" title="c19">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 <b>by </b><i><a class="txt" href="#E2:48_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2"/>
( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  &amp; not <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E47">A74</font></i>: 
<font color="Maroon" title="c14">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="glib_000.html#T74" title="GLIB_000:th.74">GLIB_000:74</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">v'</font> = <font color="Maroon" title="c14">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c17">v'</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon" title="c19">v</font> = <font color="Maroon" title="c18">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Maroon" title="c19">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>thus </b><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c19">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a></span><br/><b>consider </b><font color="Maroon" title="c20">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c19">v</font>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_2"/><i><font color="Green" title="E48">A75</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c20">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_2"/>
not <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">cv'</font> = <font color="Maroon" title="c3">cv</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c20">G3</font> <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E48">A75</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_2"/><i><font color="Green" title="E49">A76</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c17">v'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green" title="E27">Lm16</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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" title="c22">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c20">G3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> or <font color="Maroon" title="c22">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">G3</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c22">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14" title="GLIB_001:th.14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E50">A77</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> or  not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green" title="E51">A78</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E48">A75</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:48_1_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" title="E52">A79</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c16">C'R2</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><i><font color="Green" title="E51">A78</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:48_1_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E47">A74</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c16">C'R2</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c16">C'R2</font><b> such that </b><br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green" title="E53">A80</font></i>: 
<font color="Maroon" title="c23">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E44">A51</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A79</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green" title="E54">A81</font></i>: 
( <font color="Maroon" title="c23">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Maroon" title="c23">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> )
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><i><font color="Green" title="E53">A80</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><i><font color="Green" title="E55">A82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_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" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E3:48_1_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" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c24">W</font> = <font color="Maroon" title="c23">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">C</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">W</font> = <font color="Maroon" title="c24">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">W</font> = <font color="Maroon" title="c25">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><i><font color="Green" title="E55">A82</font></i></a>, <a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">W</font> = <font color="Maroon" title="c26">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green" title="E56">A83</font></i>: 
<font color="Maroon" title="c27">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A81</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E16:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="txt" href="#E2:48_1_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E47">A74</font></i></a>, <a class="ref" href="glib_000.html#T68" title="GLIB_000:th.68">GLIB_000:68</a></i>;<br/><a NAME="E17:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E18:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D42" title="GLIB_000:def.42">GLIB_000:def 42</a></i>;<br/><a NAME="E19:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T38" title="GLIB_000:th.38">GLIB_000:38</a></i>;<br/><a NAME="E20:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c20">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E21:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c20">G3</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="glib_000.html#T76" title="GLIB_000:th.76">GLIB_000:76</a></i>;<br/><a NAME="E22:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c20">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><a NAME="E23:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c27">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c3">cv</font>
 <b>by </b><i><a class="txt" href="#E15:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><i><font color="Green" title="E56">A83</font></i></a>, <a class="ref" href="glib_001.html#T67" title="GLIB_001:th.67">GLIB_001:67</a></i>;<br/><a NAME="E24:48_1_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" title="c27">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c11">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24" title="GLIB_001:th.24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E25:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green" title="E51">A84</font></i>: 
not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c23">y'</font> = <font color="Maroon" title="c22">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green" title="E52">A85</font></i>: 
<font color="Maroon" title="c24">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c25">P</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green" title="E53">A86</font></i>: 
<font color="Maroon" title="c25">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A85</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E54">A87</font></i>: 
( <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y'</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c19">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c26">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E55">A88</font></i>: 
( <font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font> &amp; <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_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" title="E56">A89</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font>
 <b>by </b><i><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A87</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E55">A88</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>set </b><font color="Maroon" title="c27">P2</font> = <font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E57">A90</font></i>: 
<font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A87</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E55">A88</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c3">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c28">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E58">A91</font></i>: 
( <font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_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" title="c28">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E58">A91</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:48_1_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" title="E59">A92</font></i>: 
( <font color="Maroon" title="c25">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">P</font> )
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E55">A88</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E58">A91</font></i></a>, <a class="ref" href="glib_001.html#T48" title="GLIB_001:th.48">GLIB_001:48</a></i>;<br/><a NAME="E7:48_1_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#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font>
 <b>by </b><i><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E56">A89</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E8:48_1_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></i>;<br/><a NAME="E9:48_1_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" title="E60">A93</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E10:48_1_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" title="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E59">A92</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:48_1_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="#E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"><i><font color="Green" title="E50">A77</font></i></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A87</font></i></a>, <a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E59">A92</font></i></a>, <a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E60">A93</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c28">P2</font> = <font color="Maroon" title="c25">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c26">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">P</font></span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c28">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c23">y'</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"><i><font color="Green" title="E57">A90</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E10:48_1_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="#E10:48_1_1_1_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E49">A76</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E51">A84</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c26">P</font> = <font color="Maroon" title="c25">P</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon" title="c27">P</font> = <font color="Maroon" title="c26">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font></span><br/><b>thus </b><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c27">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"><i><font color="Green" title="E53">A86</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_2_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">cv'</font>,<font color="Maroon" title="c22">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">G3</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E18"><i><font color="Green" title="E17">Lm6</font></i></a></i>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">G3</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green" title="E29">Lm18</font></i></a></i>;<br/><b>hence </b><a NAME="E14:48_1_1_1_1_2_1_2_1_2_1_1_2"/>
not <font color="Maroon" title="c19">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 <b>by </b><i><a class="txt" href="#E2:48_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  &amp; <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cv</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green" title="E47">A94</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c14">v2</font>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T74" title="GLIB_000:th.74">GLIB_000:74</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green" title="E48">A95</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green" title="E47">A94</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><b>set </b><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c14">v2</font>;<br/><b>reconsider </b><font color="Maroon" title="c19">v'</font> = <font color="Maroon" title="c13">v1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">v</font> = <font color="Maroon" title="c19">v'</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>take </b><font color="Maroon" title="c21">v</font> = <font color="Maroon" title="c20">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Maroon" title="c21">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>thus </b><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
<font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c21">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a></span><br/><b>consider </b><font color="Maroon" title="c22">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c21">v</font>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green" title="E49">A96</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c22">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
not <font color="Maroon" title="c3">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">cv'</font> = <font color="Maroon" title="c3">cv</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c22">G3</font> <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green" title="E49">A96</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_3"/><i><font color="Green" title="E50">A97</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c19">v'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E28"><i><font color="Green" title="E27">Lm16</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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" title="c24">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c22">G3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1"/>
( <font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> or <font color="Maroon" title="c24">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c22">G3</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c24">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14" title="GLIB_001:th.14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E51">A98</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1"/>
( <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> or  not <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green" title="E52">A99</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green" title="E49">A96</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:48_1_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" title="E53">A100</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">C'R1</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A99</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E43">A50</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">C'R1</font>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c15">C'R1</font><b> such that </b><br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green" title="E54">A101</font></i>: 
<font color="Maroon" title="c25">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c14">v2</font>
 <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E44">A51</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><i><font color="Green" title="E53">A100</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green" title="E55">A102</font></i>: 
( <font color="Maroon" title="c25">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y</font> &amp; <font color="Maroon" title="c25">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v2</font> )
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A101</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><i><font color="Green" title="E56">A103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_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" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E46">A53</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E3:48_1_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" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c26">W</font> = <font color="Maroon" title="c25">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c9">C</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">W</font> = <font color="Maroon" title="c26">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">W</font> = <font color="Maroon" title="c27">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><i><font color="Green" title="E56">A103</font></i></a>, <a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">W</font> = <font color="Maroon" title="c28">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green" title="E57">A104</font></i>: 
<font color="Maroon" title="c29">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c14">v2</font>
 <b>by </b><i><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><i><font color="Green" title="E55">A102</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E16:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E43">A50</font></i></a>, <a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green" title="E47">A94</font></i></a>, <a class="ref" href="glib_000.html#T68" title="GLIB_000:th.68">GLIB_000:68</a></i>;<br/><a NAME="E17:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green" title="E48">A95</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E18:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D42" title="GLIB_000:def.42">GLIB_000:def 42</a></i>;<br/><a NAME="E19:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T38" title="GLIB_000:th.38">GLIB_000:38</a></i>;<br/><a NAME="E20:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c22">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E21:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c14">v2</font>,<font color="Maroon" title="c22">G3</font>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green" title="E47">A94</font></i></a>, <a class="ref" href="glib_000.html#T76" title="GLIB_000:th.76">GLIB_000:76</a></i>;<br/><a NAME="E22:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c14">v2</font>,<font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c22">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>;<br/><a NAME="E23:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c29">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c17">e</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c3">cv</font>
 <b>by </b><i><a class="txt" href="#E15:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><i><font color="Green" title="E57">A104</font></i></a>, <a class="ref" href="glib_001.html#T67" title="GLIB_001:th.67">GLIB_001:67</a></i>;<br/><a NAME="E24:48_1_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" title="c29">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c24">y</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24" title="GLIB_001:th.24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E25:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green" title="E52">A105</font></i>: 
not <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c25">y'</font> = <font color="Maroon" title="c24">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green" title="E53">A106</font></i>: 
<font color="Maroon" title="c26">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c25">y'</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c27">P</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c26">W</font>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green" title="E54">A107</font></i>: 
<font color="Maroon" title="c27">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">cv</font>,<font color="Maroon" title="c25">y'</font>
 <b>by </b><i><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><i><font color="Green" title="E53">A106</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E55">A108</font></i>: 
( <font color="Maroon" title="c27">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <font color="Maroon" title="c27">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y'</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c28">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E56">A109</font></i>: 
( <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font> &amp; <font color="Maroon" title="c27">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_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" title="E57">A110</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">n</font>
 <b>by </b><i><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"><i><font color="Green" title="E56">A109</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>set </b><font color="Maroon" title="c29">P2</font> = <font color="Maroon" title="c27">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c28">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E58">A111</font></i>: 
<font color="Maroon" title="c27">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c28">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">v</font>,<font color="Maroon" title="c25">y'</font>
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"><i><font color="Green" title="E56">A109</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c3">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c28">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c30">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E59">A112</font></i>: 
( <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c28">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c27">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c28">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:48_1_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" title="c30">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c28">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E59">A112</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:48_1_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" title="E60">A113</font></i>: 
( <font color="Maroon" title="c27">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">cv</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">P</font> )
 <b>by </b><i><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"><i><font color="Green" title="E56">A109</font></i></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E59">A112</font></i></a>, <a class="ref" href="glib_001.html#T48" title="GLIB_001:th.48">GLIB_001:48</a></i>;<br/><a NAME="E7:48_1_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#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">m</font>
 <b>by </b><i><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E8:48_1_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></i>;<br/><a NAME="E9:48_1_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" title="E61">A114</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E10:48_1_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" title="c28">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E60">A113</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:48_1_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="#E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"><i><font color="Green" title="E51">A98</font></i></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E60">A113</font></i></a>, <a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E61">A114</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c30">P2</font> = <font color="Maroon" title="c27">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c28">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">P</font></span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c30">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">v</font>,<font color="Maroon" title="c25">y'</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E10:48_1_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="#E12:48_1_1_1_1_2_1_2_1_2_1_1_3"><i><font color="Green" title="E50">A97</font></i></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><i><font color="Green" title="E52">A105</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c28">P</font> = <font color="Maroon" title="c27">P</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> <b>by </b><i><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></i>;<br/><b>take </b><font color="Maroon" title="c29">P</font> = <font color="Maroon" title="c28">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c29">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font></span><br/><b>thus </b><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c29">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font>
 <b>by </b><i><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A107</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:48_1_1_1_1_2_1_2_1_2_1_1_3_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c22">G3</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">cv'</font>,<font color="Maroon" title="c24">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E14:48_1_1_1_1_2_1_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c22">G3</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E18"><i><font color="Green" title="E17">Lm6</font></i></a></i>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c22">G3</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green" title="E29">Lm18</font></i></a></i>;<br/><b>hence </b><a NAME="E16:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
not <font color="Maroon" title="c21">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 <b>by </b><i><a class="txt" href="#E2:48_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:48_1_1_1_1_2_1_2_1_2"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:48_1_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">C</font> &amp; not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c9">CC1</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c4">G2</font>,<font color="Maroon" title="c7">C1</font>;<br/><b>consider </b><font color="Maroon" title="c10">CC2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c4">G2</font>,<font color="Maroon" title="c8">C2</font>;<br/><b>consider </b><font color="Maroon" title="c11">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E21:48_1_1_1_1_2"/><i><font color="Green" title="E41">A115</font></i>: 
( <font color="Maroon" title="c11">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">CC1</font> &amp; not <font color="Maroon" title="c11">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="#E17:48_1_1_1_1_2"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E23:48_1_1_1_1_2"/><i><font color="Green" title="E42">A116</font></i>: 
( <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c10">CC2</font> &amp; not <font color="Maroon" title="c12">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="#E17:48_1_1_1_1_2"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c13">v1</font> = <font color="Maroon" title="c11">v1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v2</font> &amp; not <font color="Maroon" title="c13">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b1">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>take </b><font color="Maroon" title="c14">v2</font> = <font color="Maroon" title="c12">v2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">v2</font> &amp; not <font color="Maroon" title="c13">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c14">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_1_1_2_2"/>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E43">A117</font></i>: 
<font color="Maroon" title="c9">CC1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c10">CC2</font>
 <b>by </b><i><a class="txt" href="#E21:48_1_1_1_1_2"><i><font color="Green" title="E41">A115</font></i></a>, <a class="txt" href="#E23:48_1_1_1_1_2"><i><font color="Green" title="E42">A116</font></i></a>, <a class="txt" href="glib_002.html#E29"><i><font color="Green" title="E28">Lm17</font></i></a></i>;<br/><a NAME="E3:48_1_1_1_1_2_2"/>
( <font color="Maroon" title="c7">C1</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> &amp; <font color="Maroon" title="c8">C2</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E25"><i><font color="Green" title="E24">Lm13</font></i></a></i>;<br/><a NAME="E4:48_1_1_1_1_2_2"/><b>then </b>
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">CC1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C1</font> &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c10">CC2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C2</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E5:48_1_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:48_1_1_1_1_2"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E2:48_1_1_1_1_2_2"><i><font color="Green" title="E43">A117</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E25:48_1_1_1_1_2"/>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c13">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c14">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/><b>thus </b><a NAME="E26:48_1_1_1_1_2"/>
( not <font color="Maroon" title="c13">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c14">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="#E21:48_1_1_1_1_2"><i><font color="Green" title="E41">A115</font></i></a>, <a class="txt" href="#E23:48_1_1_1_1_2"><i><font color="Green" title="E42">A116</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_1"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:48"/><b>then </b><i><font color="Green" title="E33">A118</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<a NAME="E3:48"/><i><font color="Green" title="E34">A119</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E2:48"><i><font color="Green" title="E33">A118</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> non <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span><br/>

<a NAME="E4:48"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> 
 
;<br/>
<b>hence </b><a NAME="E5:48"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="#E3:48"><i><font color="Green" title="E34">A119</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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