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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="glib_002.html#NM1">Component</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:44_1_1_1"/><i><font color="Green">E29</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>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:44_1"><i><font color="Green">E28</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="E6:44_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:44_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E7:44_1_1_1"/>
<font color="Maroon">c<sub>4</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>3</sub></font>
 <b>by </b><i><a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:44_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</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>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <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>5</sub></font></span>)</span>;<br/><a NAME="E5:44_1_1"/>
( not <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#R7">c&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R7">c&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="glib_002.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E6:44_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( ( <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#R6">c=</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R6">c=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="glib_000.html#D38">GLIB_000:def 38</a></i>;<br/><a NAME="E7:44_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>6</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>5</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>6</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>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E8:44_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/><a NAME="E9:44_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>5</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>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:44_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="glib_002.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:44_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:44_1"><i><font color="Green">E28</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>9</sub></font> = <font color="Maroon">c<sub>8</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="E5:44_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:44_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E6:44_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:44_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="glib_002.html#E9:44_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="glib_002.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><a NAME="E11:44_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1_1_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>set </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>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>set </b><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>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E1:44_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:44_1_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:44_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E4:44_1_1_3"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</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/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:44_1_1_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:44_1"><i><font color="Green">E28</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>13</sub></font> = <font color="Maroon">c<sub>12</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="E8:44_1_1_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:44_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:44_1_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:44_1"><i><font color="Green">E28</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>15</sub></font> = <font color="Maroon">c<sub>14</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="E12:44_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E10:44_1_1_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E13:44_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>10</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>5</sub></font> &amp; <font color="Maroon">c<sub>11</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>5</sub></font> )
 <b>by </b><i><a class="ref" href="glib_002.html#D5" target="_self">Def5</a>, <a class="txt" href="glib_002.html#E9:44_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="glib_002.html#E8:44_1_1_3"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E14:44_1_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E4:44_1_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="glib_002.html#E7:44_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="glib_000.html#T35">GLIB_000:35</a></i>;<br/></div><b>end;</b></div><a NAME="E13:44_1_1"/><b>then </b>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E14:44_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E11:44_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/><b>hence </b><a NAME="E15:44_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E6:44_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="glib_002.html#E8:44_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a>, <a class="ref" href="glib_000.html#D37">GLIB_000:def 37</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:44_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</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>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:44"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</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>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E3:44"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/>


</div>
