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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>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:42"/>
<font color="Maroon">c<sub>2</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>2</sub></font></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> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T79">GLIB_000:79</a></i>;<br/>
<a NAME="E3:42"/><b>then </b><i><font color="Green">E26</font></i>: 
 <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> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:42_1"/>
( <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>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E3:42"><i><font color="Green">E26</font></i></a></i>;<br/><b>assume </b><a NAME="E2:42_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green">E28</font></i>: 
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>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E3:42_1_1"/><i><font color="Green">E29</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>3</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> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E4:42_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is  <a href="glib_000.html#V9">spanning</a>  <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#E3:42"><i><font color="Green">E26</font></i></a>, <a class="ref" href="glib_000.html#T47">GLIB_000:47</a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><a NAME="E5:42_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E22"><i><font color="Green">Lemma21</font></i></a></i>;<br/><a NAME="E6:42_1_1"/><b>then </b>
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>
 <b>by </b><i><a class="ref" href="glib_002.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E7:42_1_1"/><b>then </b>
( <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>2</sub></font> <a href="glib_000.html#R6">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D38">GLIB_000:def 38</a></i>;<br/><b>hence </b><a NAME="E8:42_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E1:42_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="glib_002.html#E2:42_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="glib_002.html#E3:42_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a>, <a class="txt" href="glib_002.html#E4:42_1_1"><i><font color="Green">E30</font></i></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="E4:42_1"/>
<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>2</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
 <a href="glib_000.html#K26">the_Edges_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_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>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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