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

<b>let </b><font color="Maroon" title="c3">G2</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>;<br/>

<a NAME="E1:24_1_1"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font>;<br/><b>consider </b><font color="Maroon" title="c6">W1R</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:24_1_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">W2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:24_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c7">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>;<br/><b>set </b><font color="Maroon" title="c8">W1</font> = <font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> ;<br/><b>set </b><font color="Maroon" title="c9">W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font>;<br/><a NAME="E5:24_1_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:24_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="glib_001.html#T24" title="GLIB_001:th.24">GLIB_001:24</a></i>;<br/><a NAME="E6:24_1_1_1"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></i>;<br/><a NAME="E7:24_1_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c7">W2</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="c2">v</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="glib_002.html#E5:24_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E8:24_1_1_1"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T94" title="GLIB_001:th.94">GLIB_001:94</a></i>;<br/><a NAME="E9:24_1_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E7:24_1_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><a NAME="E10:24_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E8:24_1_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:24_1_1_1"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="glib_002.html#E11"><i><font color="Green" title="E11">Lm4</font></i></a></i>;<br/><a NAME="E12:24_1_1_1"/><b>then </b><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T39" title="GLIB_000:th.39">GLIB_000:39</a></i>;<br/><a NAME="E13:24_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T110" title="GLIB_001:th.110">GLIB_001:110</a></i>;<br/><a NAME="E14:24_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:24_1_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1R</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c7">W2</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G2</font> <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="glib_002.html#E11:24_1_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></i>;<br/><b>take </b><font color="Maroon" title="c11">W</font> = <font color="Maroon" title="c10">W</font>;<br/><b>thus </b><a NAME="E16:24_1_1_1"/>
<font color="Maroon" title="c11">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:24_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:24_1_1"/><b>then </b><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c3">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 
<b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">G3</font> being  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E2:24_1_1_2"/><i><font color="Green" title="E15">A11</font></i>: 
<font color="Maroon" title="c3">G2</font> <a href="glib_000.html#R7" title="GLIB_000:pred.7">c&lt;</a> <font color="Maroon" title="c4">G3</font>
 ;<br/><a NAME="E3:24_1_1_2"/>
<font color="Maroon" title="c3">G2</font> <a href="glib_000.html#R6" title="GLIB_000:pred.6">c=</a> <font color="Maroon" title="c4">G3</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:24_1_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="glib_000.html#D38" title="GLIB_000:def.38">GLIB_000:def 38</a></i>;<br/><a NAME="E4:24_1_1_2"/><b>then </b><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c3">G2</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c4">G3</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></i>;<br/><a NAME="E5:24_1_1_2"/><i><font color="Green" title="E17">A13</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="glib_002.html#E8"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/><a NAME="E6:24_1_1_2"/><i><font color="Green" title="E18">A14</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>;<br/><div><i><font color="Green" title="E19">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:24_1_1_2_1"/><i><font color="Green" title="E20">A16</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c6">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G3</font><b> such that </b><br/><a NAME="E4:24_1_1_2_1"/><i><font color="Green" title="E21">A17</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E5:24_1_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="glib_002.html#E6:24_1_1_2"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="glib_002.html#E2:24_1_1_2_1"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">W</font> = <font color="Maroon" title="c6">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><a NAME="E6:24_1_1_2_1"/>
<font color="Maroon" title="c7">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_2_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E7:24_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="glib_002.html#E2:24_1_1_2_1"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></i>;<br/></div><b>end;</b></div><a NAME="E8:24_1_1_2"/><i><font color="Green" title="E20">A18</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_2_2_1"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> ) or  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">G3</font> &amp; not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c3">G2</font> ) )
 </span><b>by </b><i><a class="txt" href="glib_002.html#E2:24_1_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="glib_000.html#T102" title="GLIB_000:th.102">GLIB_000:102</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_2_2_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E7:24_1_1_2"><i><font color="Green" title="E19">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2_2_1_2"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">G3</font> &amp; not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c3">G2</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1_2_2_1_2"/><i><font color="Green" title="E21">A19</font></i>: 
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">G3</font> &amp; not <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c3">G2</font> )
 ;<br/><b>set </b><font color="Maroon" title="c6">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>;<br/><b>set </b><font color="Maroon" title="c7">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>;<br/><a NAME="E4:24_1_1_2_2_1_2"/><i><font color="Green" title="E22">A20</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">G3</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:24_1_1_2_2_1_2"><i><font color="Green" title="E21">A19</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><a NAME="E5:24_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E23">A21</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_2_2_1_2_1_1"/>
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> or  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_2_2_1_2_1_1_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c8">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>, <font color="Maroon" title="c9">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font> <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_2_2_1_2"><i><font color="Green" title="E22">A20</font></i></a>, <a class="ref" href="glib_000.html#T16" title="GLIB_000:th.16">GLIB_000:16</a></i>;<br/><a NAME="E3:24_1_1_2_2_1_2_1_1_1"/>
( <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:24_1_1_2_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="glib_002.html#E3:24_1_1_2_2_1_2"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="glib_002.html#E5:24_1_1_2_2_1_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="glib_000.html#T35" title="GLIB_000:th.35">GLIB_000:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2_2_1_2_1_1_2"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_2_1_2_1_1_2"/><b>then </b>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E8:24_1_1_2"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E7:24_1_1_2"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:24_1_1_2_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:24_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_1_1"/>
<font color="Maroon" title="c3">G2</font> is <a href="glib_002.html#V4" title="GLIB_002:attr.4">Component-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:24_1_1"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="glib_002.html#D7" target="_self" title="GLIB_002:def.7">Def7</a></i>;<br/>


</div>
