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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> 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#K15" title="GLIB_000:func.15">.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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">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="c2">k</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 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="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c3">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/><span class="kw">assume </span><a NAME="E1:48_1_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c3">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c3">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E30"><span class="lab"><font color="Green" title="E29">Lm18</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c3">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="c3">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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 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="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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"><span class="kw">consider </span><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T21" title="GLIB_000:th.21">GLIB_000:21</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">G</font> st <br/>( <font color="Maroon" title="c6">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="c6">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/><span class="kw">take </span><font color="Maroon" title="c7">v2</font> = <font color="Maroon" title="c5">v2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> &amp;  not <font color="Maroon" title="c6">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  &amp;  not <font color="Maroon" title="c7">v2</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )</span><br/><span class="kw">thus </span><a NAME="E4:48_1_1_1_1_1"/>
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  &amp;  not <font color="Maroon" title="c7">v2</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )</span><br/><span class="kw">thus </span><a NAME="E5:48_1_1_1_1_1"/>
(  not <font color="Maroon" title="c6">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  &amp;  not <font color="Maroon" title="c7">v2</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2"><span class="kw">suppose </span></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="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">cv</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c4">cv</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 ;<br/><span class="kw">set </span><font color="Maroon" title="c5">G2</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>;<br/><a NAME="E4:48_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></span>;<br/><a NAME="E5:48_1_1_1_1_2"/><span class="kw">then </span>
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>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:48_1_1_1_1_2"/><span class="kw">then </span>
 <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"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/><a NAME="E7:48_1_1_1_1_2"/><span class="kw">then </span>
 <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"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/><a NAME="E8:48_1_1_1_1_2"/><span class="kw">then </span>
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"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">C1</font>, <font color="Maroon" title="c7">C2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
( <font color="Maroon" title="c6">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Maroon" title="c7">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  )
 <span class="kw">and </span><br/><a NAME="E11:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c6">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">C1</font> = <font color="Maroon" title="c6">C1</font>, <font color="Maroon" title="c9">C2</font> = <font color="Maroon" title="c7">C2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">CC1</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c8">C1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">CC2</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c9">C2</font>;<br/><a NAME="E13:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</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="c3">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="c4">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E14:48_1_1_1_1_2"/>
<font color="Maroon" title="c3">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.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="c3">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="c4">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="c3">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">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T35" title="GLIB_000:th.35">GLIB_000:35</a></span>;<br/><a NAME="E15:48_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.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="c3">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="c4">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="c3">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="c4">cv</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D40" title="GLIB_000:def.40">GLIB_000:def 40</a></span>;<br/><a NAME="E16:48_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A10</font></span>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</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="c3">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="c4">cv</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E17:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.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="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="glib_000.html#T48" title="GLIB_000:th.48">GLIB_000:48</a></span>;<br/><div><span class="lab"><font color="Green" title="E42">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">C</font> be   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">x</font> =  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font>;<br/><span class="kw">assume </span><a NAME="E1:48_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
 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="c12">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="c4">cv</font>,<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_1_1_2_1_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">x9</font> =  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E4:48_1_1_1_1_2_1_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</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="c3">G</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x99</font> = <font color="Maroon" title="c14">x9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E7:48_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c15">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font>;<br/><a NAME="E8:48_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c15">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="glib_001.html#T160" title="GLIB_001:th.160">GLIB_001:160</a></span>;<br/><a NAME="E9:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A16</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>);<br/><a NAME="E10:48_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E47">A17</font></span>: 
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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A19</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></span>;<br/><a NAME="E13:48_1_1_1_1_2_1_1"/>
 not <font color="Maroon" title="c14">x9</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="c4">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E14:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A20</font></span>: 
 the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E15:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A21</font></span>: 
 not  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></span>;<br/><a NAME="E16:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T125" title="GLIB_001:th.125">GLIB_001:125</a></span>;<br/><a NAME="E17:48_1_1_1_1_2_1_1"/><span class="kw">then </span>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3, the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><a NAME="E18:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A23</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3, the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></span>;<br/><a NAME="E19:48_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E54">A24</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">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> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E55">A25</font></span>: 
<font color="Maroon" title="c19">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><a NAME="E5:48_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E57">A27</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E6:48_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A28</font></span>: 
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="c19">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:48_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A29</font></span>: 
<font color="Maroon" title="c19">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E9:48_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A30</font></span>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E10:48_1_1_1_1_2_1_1_1"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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="c19">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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="txt" href="#E7:48_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><span class="kw">hence </span><a NAME="E11:48_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E19:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="txt" href="#E6:48_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><a NAME="E22:48_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c19">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3, the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><a NAME="E23:48_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">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>  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font>, the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></span>;<br/><a NAME="E24:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A31</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c14">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></span>;<br/><a NAME="E25:48_1_1_1_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="ref" href="glib_001.html#T125" title="GLIB_001:th.125">GLIB_001:125</a></span>;<br/><a NAME="E26:48_1_1_1_1_2_1_1"/><span class="kw">then </span>
<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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E27:48_1_1_1_1_2_1_1"/><span class="kw">then </span>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E28:48_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A32</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</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="c4">cv</font>, the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c16">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></span>;<br/><a NAME="E29:48_1_1_1_1_2_1_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c14">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E28"><span class="lab"><font color="Green" title="E27">Lm16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E30:48_1_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E28:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E24:48_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font>, <font color="Maroon" title="c14">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A33</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c3">G</font>
 ;<br/><a NAME="E4:48_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A34</font></span>: 
<font color="Maroon" title="c14">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="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_1_2_1_2_1"/>
( <font color="Maroon" title="c12">C</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  or  not <font color="Maroon" title="c12">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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c12">C</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">v99</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c12">C</font><span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E45">A35</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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="c15">v99</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T22" title="GLIB_000:th.22">GLIB_000:22</a></span>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">v9</font> = <font color="Maroon" title="c15">v99</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">v</font> = <font color="Maroon" title="c16">v9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c17">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> &amp;  not <font color="Maroon" title="c18">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )</span><br/><span class="kw">thus </span><a NAME="E7:48_1_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c18">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c18">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> </span><br/><a NAME="E8:48_1_1_1_1_2_1_2_1_1"/>
 not <font color="Maroon" title="c16">v9</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="c4">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A36</font></span>: 
<font color="Maroon" title="c16">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E47">A37</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v9</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>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c16">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E45">A35</font></span></a>, <a class="txt" href="glib_002.html#E28"><span class="lab"><font color="Green" title="E27">Lm16</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c19">G3</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</font>;<br/><div><span class="lab"><font color="Green" title="E48">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">e</font>, <font color="Maroon" title="c21">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c20">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c21">z</font>,<font color="Maroon" title="c3">G</font> or <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> or <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font> )</span><br/><span class="kw">assume </span><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A39</font></span>: 
<font color="Maroon" title="c20">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c21">z</font>,<font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> or <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font> )</span><br/><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A40</font></span>: 
<font color="Maroon" title="c20">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="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A41</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">v</font>
 <span class="kw">and </span><br/><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A42</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_1_1"/>
 not <font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E46">A36</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E49">A39</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_2_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="glib_000.html#T65" title="GLIB_000:th.65">GLIB_000:65</a></span>;<br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E50">A40</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c21">z</font>, the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E49">A39</font></span></a>, <a class="ref" href="glib_000.html#T73" title="GLIB_000:th.73">GLIB_000:73</a></span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c16">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E8"><span class="lab"><font color="Green" title="E8">Lm1</font></span></a>, <a class="txt" href="glib_002.html#E9"><span class="lab"><font color="Green" title="E9">Lm2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_1_1_1_2_1_2_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_1"/>
( <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> or <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c21">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</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="c20">x</font>,<font color="Maroon" title="c21">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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> or <font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E49">A43</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</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="c20">x</font>,<font color="Maroon" title="c21">y</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c22">W</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c20">x</font>;<br/><span class="kw">take </span><font color="Maroon" title="c23">W</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c20">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c20">x</font>,<font color="Maroon" title="c21">y</font></span><br/><span class="kw">thus </span><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_2_1_1_1"/>
<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="c20">x</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A43</font></span></a>, <a class="ref" href="glib_001.html#T13" title="GLIB_001:th.13">GLIB_001:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A44</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</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="c20">x</font>,<font color="Maroon" title="c21">y</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c22">x9</font> = <font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c23">y9</font> = <font color="Maroon" title="c21">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">consider </span><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="c3">G</font><span class="kw"> such that </span><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E50">A45</font></span>: 
<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="c22">x9</font>,<font color="Maroon" title="c23">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c25">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E51">A46</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <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="c22">x9</font>,<font color="Maroon" title="c23">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E50">A45</font></span></a>, <a class="ref" href="glib_001.html#T160" title="GLIB_001:th.160">GLIB_001:160</a></span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E52">A47</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</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="c3">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="c18">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c20">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="c18">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A48</font></span>: 
<font color="Maroon" title="c18">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/>
 not <font color="Maroon" title="c21">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="c18">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A47</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A49</font></span>: 
<font color="Maroon" title="c18">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></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_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c18">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E55">A50</font></span>: 
<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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E56">A51</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</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="c18">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><a NAME="E5:48_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="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A48</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span>
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="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">n2</font> = <font color="Maroon" title="c26">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><div><span class="lab"><font color="Green" title="E57">A52</font></span>: <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"><span class="kw">now </span></a><div class="add"><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E58">A53</font></span>: 
<font color="Maroon" title="c27">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><span class="kw">assume </span><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A48</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E55">A50</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E58">A53</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c28">e1</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c29">e2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 1</span>)</span>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c26">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E55">A50</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n2</font>, the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A54</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A38</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E57">A52</font></span></a>, <a class="ref" href="glib_000.html#T14" title="GLIB_000:th.14">GLIB_000:14</a></span>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A55</font></span>: 
<font color="Maroon" title="c26">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A49</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E55">A50</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E14:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A56</font></span>: 
<font color="Maroon" title="c26">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T1" title="GLIB_001:th.1">GLIB_001:1</a></span>;<br/><div><span class="lab"><font color="Green" title="E61">A57</font></span>: <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"><span class="kw">now </span></a><div class="add"><a NAME="E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E62">A58</font></span>: 
<font color="Maroon" title="c26">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="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">assume </span><a NAME="E2:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A48</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="txt" href="#E14:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E16:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c27">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E17:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A59</font></span>: 
<font color="Maroon" title="c27">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="c26">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E18:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">v</font>, the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 2</span>)</span>,<font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E19:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A60</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A38</font></span></a>, <a class="txt" href="#E15:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E61">A57</font></span></a></span>;<br/><a NAME="E20:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="txt" href="#E14:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E17:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E62">A59</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E21:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A61</font></span>: 
<font color="Maroon" title="c4">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E22:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="txt" href="#E14:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E19:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E63">A60</font></span></a>, <a class="txt" href="#E17:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E62">A59</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E23:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A44</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="txt" href="#E21:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E64">A61</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c24">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c27">P</font> = <font color="Maroon" title="c26">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c20">x</font>,<font color="Maroon" title="c21">y</font></span><br/><span class="kw">thus </span><a NAME="E15:48_1_1_1_1_2_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="c20">x</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</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="c20">x</font>,<font color="Maroon" title="c21">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:48_1_1_1_1_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A62</font></span>: 
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><span class="kw">assume </span><a NAME="E4:48_1_1_1_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c18">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:48_1_1_1_1_2_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c18">v</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_1_1_1_2_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A62</font></span></a>, <a class="txt" href="glib_002.html#E30"><span class="lab"><font color="Green" title="E29">Lm18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:48_1_1_1_1_2_1_2_1_1"/>
 not <font color="Maroon" title="c18">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2"/>
 not <font color="Maroon" title="c12">C</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">C9</font> = <font color="Maroon" title="c12">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"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">C</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.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"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">v1</font>, <font color="Maroon" title="c17">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c15">C9</font><span class="kw"> such that </span><br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E45">A63</font></span>: 
<font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">v2</font>
 <span class="kw">and </span><br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E46">A64</font></span>: 
 not <font color="Maroon" title="c16">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 <span class="kw">and </span><br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E47">A65</font></span>: 
 not <font color="Maroon" title="c17">v2</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E17:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">C9R1</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E48">A66</font></span>: 
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E46">A64</font></span></a>, <a class="txt" href="glib_002.html#E32"><span class="lab"><font color="Green" title="E31">Lm20</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">C9R2</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c17">v2</font>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E49">A67</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</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="c15">C9</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="c16">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c17">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c17">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="c4">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A68</font></span>: 
<font color="Maroon" title="c17">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E51">A69</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c17">v2</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="c15">C9</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="c17">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E14:48_1_1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c16">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="c4">cv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E16:48_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A70</font></span>: 
<font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E17:48_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E53">A71</font></span>: 
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c17">v2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E47">A65</font></span></a>, <a class="txt" href="glib_002.html#E32"><span class="lab"><font color="Green" title="E31">Lm20</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1"/>
(  not <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  or ( <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  &amp;  not <font color="Maroon" title="c17">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  ) or ( <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  &amp; <font color="Maroon" title="c17">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E54">A72</font></span>: 
 not <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">reconsider </span><font color="Maroon" title="c20">v9</font> = <font color="Maroon" title="c16">v1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">v</font> = <font color="Maroon" title="c20">v9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">v</font> = <font color="Maroon" title="c21">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> &amp;  not <font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )</span><br/><span class="kw">thus </span><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c23">G3</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E55">A73</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c3">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="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_1"/>
 not <font color="Maroon" title="c4">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="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">cv9</font> = <font color="Maroon" title="c4">cv</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E55">A73</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E56">A74</font></span>: 
<font color="Maroon" title="c16">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E54">A72</font></span></a>, <a class="ref" href="glib_000.html#T71" title="GLIB_000:th.71">GLIB_000:71</a></span>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E57">A75</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c20">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E28"><span class="lab"><font color="Green" title="E27">Lm16</font></span></a></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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font> or <font color="Maroon" title="c25">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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"><span class="kw">suppose </span></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="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c25">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T13" title="GLIB_001:th.13">GLIB_001:13</a></span>;<br/><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A76</font></span>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> or  not <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A77</font></span>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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="c13">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="c16">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A74</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A78</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<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="c25">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="c16">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E55">A73</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></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"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font><span class="kw"> such that </span><br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A79</font></span>: 
<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="c25">y</font>,<font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E48">A66</font></span></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"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A80</font></span>: 
( <font color="Maroon" title="c26">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="c25">y</font> &amp; <font color="Maroon" title="c26">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="c26">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> )
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A79</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E63">A81</font></span>: <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"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c22">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
 not <font color="Maroon" title="c22">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="c16">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
 not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A74</font></span></a>, <a class="ref" href="glib_000.html#T65" title="GLIB_000:th.65">GLIB_000:65</a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c3">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="c22">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c3">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">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D40" title="GLIB_000:def.40">GLIB_000:def 40</a></span>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.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="c3">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="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T35" title="GLIB_000:th.35">GLIB_000:35</a></span>;<br/><a NAME="E14:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c13">a</font>, the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="glib_000.html#T73" title="GLIB_000:th.73">GLIB_000:73</a></span>;<br/><a NAME="E16:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A82</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c4">cv</font>, the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T14" title="GLIB_000:th.14">GLIB_000:14</a></span>;<br/><span class="kw">reconsider </span><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="c12">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><span class="kw">reconsider </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><span class="kw">reconsider </span><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="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><a NAME="E20:48_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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E63">A81</font></span></a>, <a class="ref" href="glib_001.html#T98" title="GLIB_001:th.98">GLIB_001:98</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">W</font> = <font color="Maroon" title="c29">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><a NAME="E22:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c30">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E62">A80</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E23:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c30">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A82</font></span></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></span>;<br/><a NAME="E24:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c30">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">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="c4">cv</font>,<font color="Maroon" title="c25">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></span>;<br/><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E59">A83</font></span>: 
 not <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c26">y9</font> = <font color="Maroon" title="c25">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c27">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E60">A84</font></span>: 
<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="c4">cv</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c28">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A85</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <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="c4">cv</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E60">A84</font></span></a>, <a class="ref" href="glib_001.html#T160" title="GLIB_001:th.160">GLIB_001:160</a></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A86</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E63">A87</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A85</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></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_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">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> <span class="kw"> such that </span><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"/><span class="lab"><font color="Green" title="E64">A88</font></span>: 
<font color="Maroon" title="c29">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>
 <span class="kw">and </span><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"/><span class="lab"><font color="Green" title="E65">A89</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c30">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>);<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"/><span class="lab"><font color="Green" title="E66">A90</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">v</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E62">A86</font></span></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"><span class="lab"><font color="Green" title="E64">A88</font></span></a>, <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_1"><span class="lab"><font color="Green" title="E65">A89</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></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"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A91</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E52">A70</font></span></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"><span class="lab"><font color="Green" title="E63">A87</font></span></a>, <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_1"><span class="lab"><font color="Green" title="E65">A89</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></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_1_2_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c4">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">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> <span class="kw"> such that </span><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"/><span class="lab"><font color="Green" title="E68">A92</font></span>: 
<font color="Maroon" title="c31">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><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"/><span class="lab"><font color="Green" title="E69">A93</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<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"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></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_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A94</font></span>: 
<font color="Maroon" title="c31">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E68">A92</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A95</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E64">A88</font></span></a>, <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_1_1"><span class="lab"><font color="Green" title="E69">A93</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<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"/>
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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">m</font>
 <span class="kw">by</span> <span class="lab"><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_1"><span class="lab"><font color="Green" title="E67">A91</font></span></a>, <a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<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"/><span class="kw">then </span>
<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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<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="kw">then </span><span class="lab"><font color="Green" title="E72">A96</font></span>: 
<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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><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"/><span class="lab"><font color="Green" title="E73">A97</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E64">A88</font></span></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"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A76</font></span></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"><span class="lab"><font color="Green" title="E63">A87</font></span></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"><span class="lab"><font color="Green" title="E62">A86</font></span></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_1_1"><span class="lab"><font color="Green" title="E71">A95</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A97</font></span></a>, <a class="txt" href="#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="lab"><font color="Green" title="E72">A96</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<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"/>
<font color="Maroon" title="c31">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">v</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E66">A90</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><span class="kw">hence </span><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_1_1_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E57">A75</font></span></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"><span class="lab"><font color="Green" title="E59">A83</font></span></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c30">P</font> = <font color="Maroon" title="c29">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">cv9</font>,<font color="Maroon" title="c25">y</font></span><br/><span class="kw">thus </span><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="c30">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A85</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E18"><span class="lab"><font color="Green" title="E17">Lm6</font></span></a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E30"><span class="lab"><font color="Green" title="E29">Lm18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_1"/>
 not <font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E54">A98</font></span>: 
( <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  &amp;  not <font color="Maroon" title="c17">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">reconsider </span><font color="Maroon" title="c20">v9</font> = <font color="Maroon" title="c17">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">v</font> = <font color="Maroon" title="c20">v9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">v</font> = <font color="Maroon" title="c21">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> &amp;  not <font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )</span><br/><span class="kw">thus </span><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c23">G3</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E55">A99</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c3">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="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_2"/>
 not <font color="Maroon" title="c4">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="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E50">A68</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">cv9</font> = <font color="Maroon" title="c4">cv</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A100</font></span>: 
<font color="Maroon" title="c17">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A98</font></span></a>, <a class="ref" href="glib_000.html#T71" title="GLIB_000:th.71">GLIB_000:71</a></span>;<br/><a NAME="E9:48_1_1_1_1_2_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A101</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c20">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E28"><span class="lab"><font color="Green" title="E27">Lm16</font></span></a></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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font> or <font color="Maroon" title="c25">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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"><span class="kw">suppose </span></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="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c25">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T13" title="GLIB_001:th.13">GLIB_001:13</a></span>;<br/><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A102</font></span>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> or  not <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A103</font></span>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">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="c13">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="c17">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A104</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c17">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<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="c25">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="c17">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c17">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E51">A69</font></span></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"><span class="lab"><font color="Green" title="E59">A103</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c17">v2</font><span class="kw"> such that </span><br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A105</font></span>: 
<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="c25">y</font>,<font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E53">A71</font></span></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"><span class="lab"><font color="Green" title="E60">A104</font></span></a>, <a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A106</font></span>: 
( <font color="Maroon" title="c26">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="c25">y</font> &amp; <font color="Maroon" title="c26">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="c26">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> )
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A105</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E63">A107</font></span>: <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"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c22">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
 not <font color="Maroon" title="c22">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="c17">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
 not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E50">A68</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="ref" href="glib_000.html#T65" title="GLIB_000:th.65">GLIB_000:65</a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c3">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="c22">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c3">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">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D40" title="GLIB_000:def.40">GLIB_000:def 40</a></span>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.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="c3">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="c22">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T35" title="GLIB_000:th.35">GLIB_000:35</a></span>;<br/><a NAME="E14:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c13">a</font>, the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="glib_000.html#T73" title="GLIB_000:th.73">GLIB_000:73</a></span>;<br/><a NAME="E16:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A108</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c4">cv</font>, the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T14" title="GLIB_000:th.14">GLIB_000:14</a></span>;<br/><span class="kw">reconsider </span><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="c12">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><span class="kw">reconsider </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><span class="kw">reconsider </span><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="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><a NAME="E20:48_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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E63">A107</font></span></a>, <a class="ref" href="glib_001.html#T98" title="GLIB_001:th.98">GLIB_001:98</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">W</font> = <font color="Maroon" title="c29">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><a NAME="E22:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c30">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E62">A106</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E23:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c30">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A108</font></span></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></span>;<br/><a NAME="E24:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c30">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">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="c4">cv</font>,<font color="Maroon" title="c25">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></span>;<br/><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E59">A109</font></span>: 
 not <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c26">y9</font> = <font color="Maroon" title="c25">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c27">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E60">A110</font></span>: 
<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="c4">cv</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c28">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A111</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <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="c4">cv</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="ref" href="glib_001.html#T160" title="GLIB_001:th.160">GLIB_001:160</a></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A112</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E63">A113</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A111</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></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_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">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> <span class="kw"> such that </span><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"/><span class="lab"><font color="Green" title="E64">A114</font></span>: 
<font color="Maroon" title="c29">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>
 <span class="kw">and </span><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"/><span class="lab"><font color="Green" title="E65">A115</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c30">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>);<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"/><span class="lab"><font color="Green" title="E66">A116</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">v</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E62">A112</font></span></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"><span class="lab"><font color="Green" title="E64">A114</font></span></a>, <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_1"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></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"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A117</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E50">A68</font></span></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"><span class="lab"><font color="Green" title="E63">A113</font></span></a>, <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_1"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></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_1_2_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c4">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">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> <span class="kw"> such that </span><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"/><span class="lab"><font color="Green" title="E68">A118</font></span>: 
<font color="Maroon" title="c31">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><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"/><span class="lab"><font color="Green" title="E69">A119</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<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"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></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_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A120</font></span>: 
<font color="Maroon" title="c31">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E68">A118</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A121</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E64">A114</font></span></a>, <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_1_1"><span class="lab"><font color="Green" title="E69">A119</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<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"/>
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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">m</font>
 <span class="kw">by</span> <span class="lab"><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_1"><span class="lab"><font color="Green" title="E67">A117</font></span></a>, <a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<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"/><span class="kw">then </span>
<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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<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="kw">then </span><span class="lab"><font color="Green" title="E72">A122</font></span>: 
<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="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><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"/><span class="lab"><font color="Green" title="E73">A123</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E64">A114</font></span></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"><span class="lab"><font color="Green" title="E70">A120</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c29">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E58">A102</font></span></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"><span class="lab"><font color="Green" title="E63">A113</font></span></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"><span class="lab"><font color="Green" title="E62">A112</font></span></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_1_1"><span class="lab"><font color="Green" title="E71">A121</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A123</font></span></a>, <a class="txt" href="#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="lab"><font color="Green" title="E72">A122</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c29">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<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"/>
<font color="Maroon" title="c31">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">v</font>,<font color="Maroon" title="c26">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E66">A116</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><span class="kw">hence </span><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></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"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c27">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c30">P</font> = <font color="Maroon" title="c29">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">cv9</font>,<font color="Maroon" title="c25">y</font></span><br/><span class="kw">thus </span><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="c30">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A111</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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="c24">cv9</font>,<font color="Maroon" title="c25">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_2"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E18"><span class="lab"><font color="Green" title="E17">Lm6</font></span></a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_2"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c22">v</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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E30"><span class="lab"><font color="Green" title="E29">Lm18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_2"/>
 not <font color="Maroon" title="c22">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
( <font color="Maroon" title="c16">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  &amp; <font color="Maroon" title="c17">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">cv</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E54">A124</font></span>: 
<font color="Maroon" title="c20">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c17">v2</font>,<font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T71" title="GLIB_000:th.71">GLIB_000:71</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">v9</font> = <font color="Maroon" title="c16">v1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c17">v2</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">v</font> = <font color="Maroon" title="c21">v9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c24">v</font> = <font color="Maroon" title="c23">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> &amp;  not <font color="Maroon" title="c24">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )</span><br/><span class="kw">thus </span><a NAME="E6:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
<font color="Maroon" title="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c24">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c25">G3</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E55">A125</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c3">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="c24">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
 not <font color="Maroon" title="c4">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="c24">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">cv9</font> = <font color="Maroon" title="c4">cv</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E55">A125</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E56">A126</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c21">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E28"><span class="lab"><font color="Green" title="E27">Lm16</font></span></a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E57">A127</font></span>: 
<font color="Maroon" title="c20">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="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A124</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font> or <font color="Maroon" title="c27">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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"><span class="kw">suppose </span></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="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">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"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c27">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c26">cv9</font>,<font color="Maroon" title="c27">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T13" title="GLIB_001:th.13">GLIB_001:13</a></span>;<br/><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A128</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">cv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font> or  not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A129</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">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="c17">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="c16">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A130</font></span>: 
<font color="Maroon" title="c17">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<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="c27">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="c16">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E55">A125</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></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"><span class="lab"><font color="Green" title="E59">A129</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c15">C9</font>,<font color="Maroon" title="c16">v1</font><span class="kw"> such that </span><br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A131</font></span>: 
<font color="Maroon" title="c28">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c27">y</font>,<font color="Maroon" title="c17">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E48">A66</font></span></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"><span class="lab"><font color="Green" title="E60">A130</font></span></a>, <a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><a NAME="E8:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A132</font></span>: 
( <font color="Maroon" title="c28">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="c27">y</font> &amp; <font color="Maroon" title="c28">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="c28">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v2</font> )
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A131</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E63">A133</font></span>: <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"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c24">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
 not <font color="Maroon" title="c24">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="c16">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
 not <font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A124</font></span></a>, <a class="ref" href="glib_000.html#T65" title="GLIB_000:th.65">GLIB_000:65</a></span>;<br/><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">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="c3">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="c24">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E57">A127</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">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="c3">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">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D40" title="GLIB_000:def.40">GLIB_000:def 40</a></span>;<br/><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.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="c3">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="c24">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T35" title="GLIB_000:th.35">GLIB_000:35</a></span>;<br/><a NAME="E14:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></span>;<br/><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c17">v2</font>, the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A124</font></span></a>, <a class="ref" href="glib_000.html#T73" title="GLIB_000:th.73">GLIB_000:73</a></span>;<br/><a NAME="E16:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A134</font></span>: 
<font color="Maroon" title="c20">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">v2</font>,<font color="Maroon" title="c4">cv</font>, the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T14" title="GLIB_000:th.14">GLIB_000:14</a></span>;<br/><span class="kw">reconsider </span><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="c12">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">W</font> = <font color="Maroon" title="c29">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">W</font> = <font color="Maroon" title="c30">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><a NAME="E20:48_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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E63">A133</font></span></a>, <a class="ref" href="glib_001.html#T98" title="GLIB_001:th.98">GLIB_001:98</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">W</font> = <font color="Maroon" title="c31">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><a NAME="E22:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c32">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c27">y</font>,<font color="Maroon" title="c17">v2</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E62">A132</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E23:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c32">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c20">e</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c27">y</font>,<font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A134</font></span></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></span>;<br/><a NAME="E24:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c32">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c20">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="c4">cv</font>,<font color="Maroon" title="c27">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></span>;<br/><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></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"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E59">A135</font></span>: 
 not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">y</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c28">y9</font> = <font color="Maroon" title="c27">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c29">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E4:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E60">A136</font></span>: 
<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="c4">cv</font>,<font color="Maroon" title="c28">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c30">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font>;<br/><a NAME="E5:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A137</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <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="c4">cv</font>,<font color="Maroon" title="c28">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E60">A136</font></span></a>, <a class="ref" href="glib_001.html#T160" title="GLIB_001:th.160">GLIB_001:160</a></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A138</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E7:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E63">A139</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A137</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></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_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">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> <span class="kw"> such that </span><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"/><span class="lab"><font color="Green" title="E64">A140</font></span>: 
<font color="Maroon" title="c31">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font>
 <span class="kw">and </span><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"/><span class="lab"><font color="Green" title="E65">A141</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c32">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c31">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font></span>)</span>);<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"/><span class="lab"><font color="Green" title="E66">A142</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c31">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">v</font>,<font color="Maroon" title="c28">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E62">A138</font></span></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"><span class="lab"><font color="Green" title="E64">A140</font></span></a>, <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_1"><span class="lab"><font color="Green" title="E65">A141</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></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"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A143</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E52">A70</font></span></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"><span class="lab"><font color="Green" title="E63">A139</font></span></a>, <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_1"><span class="lab"><font color="Green" title="E65">A141</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></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_1_2_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="c4">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c31">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">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> <span class="kw"> such that </span><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"/><span class="lab"><font color="Green" title="E68">A144</font></span>: 
<font color="Maroon" title="c33">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c31">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><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"/><span class="lab"><font color="Green" title="E69">A145</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c31">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<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"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></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_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A146</font></span>: 
<font color="Maroon" title="c33">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"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c31">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E68">A144</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A147</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</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="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">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="c4">cv</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E64">A140</font></span></a>, <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_1_1"><span class="lab"><font color="Green" title="E69">A145</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<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"/>
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="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">m</font>
 <span class="kw">by</span> <span class="lab"><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_1"><span class="lab"><font color="Green" title="E67">A143</font></span></a>, <a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<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"/><span class="kw">then </span>
<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="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<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="kw">then </span><span class="lab"><font color="Green" title="E72">A148</font></span>: 
<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="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><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"/><span class="lab"><font color="Green" title="E73">A149</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E64">A140</font></span></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"><span class="lab"><font color="Green" title="E70">A146</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E12:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">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>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E58">A128</font></span></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"><span class="lab"><font color="Green" title="E63">A139</font></span></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"><span class="lab"><font color="Green" title="E62">A138</font></span></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_1_1"><span class="lab"><font color="Green" title="E71">A147</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A149</font></span></a>, <a class="txt" href="#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="lab"><font color="Green" title="E72">A148</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c31">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<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"/>
<font color="Maroon" title="c33">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">v</font>,<font color="Maroon" title="c28">y9</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E66">A142</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><span class="kw">hence </span><a NAME="E11:48_1_1_1_1_2_1_2_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1_1_1_2_1_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A126</font></span></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"><span class="lab"><font color="Green" title="E59">A135</font></span></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c29">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c32">P</font> = <font color="Maroon" title="c31">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c32">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c26">cv9</font>,<font color="Maroon" title="c27">y</font></span><br/><span class="kw">thus </span><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="c32">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c26">cv9</font>,<font color="Maroon" title="c27">y</font>
 <span class="kw">by</span> <span class="lab"><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"><span class="lab"><font color="Green" title="E61">A137</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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="c26">cv9</font>,<font color="Maroon" title="c27">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:48_1_1_1_1_2_1_2_1_2_1_1_3"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E18"><span class="lab"><font color="Green" title="E17">Lm6</font></span></a></span>;<br/><a NAME="E14:48_1_1_1_1_2_1_2_1_2_1_1_3"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c24">v</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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E30"><span class="lab"><font color="Green" title="E29">Lm18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:48_1_1_1_1_2_1_2_1_2_1_1_3"/>
 not <font color="Maroon" title="c24">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_002.html#D11" target="_self" title="GLIB_002:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E19:48_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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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="c3">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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E20:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A150</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c8">C1</font>
 <span class="kw">and </span><br/><a NAME="E21:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A151</font></span>: 
 not <font color="Maroon" title="c12">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c13">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E23:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A152</font></span>: 
<font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c9">C2</font>
 <span class="kw">and </span><br/><a NAME="E24:48_1_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A153</font></span>: 
 not <font color="Maroon" title="c13">v2</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">v1</font> = <font color="Maroon" title="c12">v1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">G</font> st <br/>( <font color="Maroon" title="c14">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="c14">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/><span class="kw">take </span><font color="Maroon" title="c15">v2</font> = <font color="Maroon" title="c13">v2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">v2</font> &amp;  not <font color="Maroon" title="c14">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  &amp;  not <font color="Maroon" title="c15">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"><span class="kw">now </span></a><div class="add"><a NAME="E1:48_1_1_1_1_2_2"/>
<font color="Maroon" title="c9">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#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E25"><span class="lab"><font color="Green" title="E24">Lm13</font></span></a></span>;<br/><a NAME="E2:48_1_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A154</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c9">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><a NAME="E3:48_1_1_1_1_2_2"/>
<font color="Maroon" title="c8">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#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E25"><span class="lab"><font color="Green" title="E24">Lm13</font></span></a></span>;<br/><a NAME="E4:48_1_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A155</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c8">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><span class="kw">assume </span><a NAME="E5:48_1_1_1_1_2_2"/>
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:48_1_1_1_1_2_2"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c8">C1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c4">cv</font>,<font color="Maroon" title="c9">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A150</font></span></a>, <a class="txt" href="#E23:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A152</font></span></a>, <a class="txt" href="glib_002.html#E29"><span class="lab"><font color="Green" title="E28">Lm17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_1_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E48">A155</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E47">A154</font></span></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E26:48_1_1_1_1_2"/>
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  &amp;  not <font color="Maroon" title="c15">v2</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )</span><br/><span class="kw">thus </span><a NAME="E27:48_1_1_1_1_2"/>
(  not <font color="Maroon" title="c14">v1</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  &amp;  not <font color="Maroon" title="c15">v2</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A151</font></span></a>, <a class="txt" href="#E24:48_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A153</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><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="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:48_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A156</font></span>: 
 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"/><span class="lab"><font color="Green" title="E34">A157</font></span>: 
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E33">A156</font></span></a>);<br/></span>
<a NAME="E4:48"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a> 
 
;<br/>
<span class="kw">hence </span><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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E34">A157</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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