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<div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<a NAME="E1:24_1_1"/><i><font color="Green">E13</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:24_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:24_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E5:24_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:24_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/><a NAME="E6:24_1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><a NAME="E7:24_1_1_1"/><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="glib_002.html#E5:24_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E8:24_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T94">GLIB_001:94</a></i>;<br/><a NAME="E9:24_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E7:24_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="glib_001.html#T89">GLIB_001:89</a></i>;<br/><a NAME="E10:24_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E8:24_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:24_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="glib_002.html#E11"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E12:24_1_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T39">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">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="glib_001.html#K14">.edges()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T110">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">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="glib_001.html#K14">.edges()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E12:24_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="glib_002.html#E11:24_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="glib_001.html#T171">GLIB_001:171</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E16:24_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:24_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:24_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 
<b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/>
<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">c<sub>4</sub></font> being  <a href="glib_002.html#V1" target="_self">connected</a>  <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:24_1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#R7">c&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:24_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#R6">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="glib_000.html#D38">GLIB_000:def 38</a></i>;<br/><a NAME="E3:24_1_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#D37">GLIB_000:def 37</a></i>;<br/><a NAME="E4:24_1_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="glib_002.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/><a NAME="E5:24_1_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:24_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><div><i><font color="Green">E19</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">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:24_1_1_2_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:24_1_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E5:24_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="glib_002.html#E4:24_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="glib_002.html#E1:24_1_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E5:24_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:24_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E6:24_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="glib_002.html#E1:24_1_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><a NAME="E7:24_1_1_2"/><i><font color="Green">E20</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:24_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="glib_000.html#D34">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="glib_000.html#T102">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></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#E6:24_1_1_2"><i><font color="Green">E19</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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1_2_2_1_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E4:24_1_1_2_2_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:24_1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E5:24_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75">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">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></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">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E4:24_1_1_2_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="glib_000.html#T16">GLIB_000:16</a></i>;<br/><a NAME="E3:24_1_1_2_2_1_2_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:24_1_1"><i><font color="Green">E13</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">E13</font></i></a>, <a class="txt" href="glib_002.html#E3:24_1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="glib_002.html#E5:24_1_1_2_2_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="glib_000.html#T35">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">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></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">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:24_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D8">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#E6:24_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rlsub_2.html#T77">RLSUB_2:77</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="E9:24_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="glib_002.html#V4" target="_self">Component-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:24_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="glib_002.html#D7" target="_self">Def7</a></i>;<br/>


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