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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="glib_000.html#M3" target="_self">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>





<b>assume </b><a NAME="E1:188"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5" target="_self">==</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="188_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="188_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:188_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> ) or  not <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  not <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="188_1_1_1"><b>suppose </b></a><a NAME="E1:188_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:188_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E1:188"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_000.html#D36" target="_self">Def36</a>, <a class="ref" href="glib_000.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:188_1_1_1"/><i><font color="Green">E81</font></i>: 
(  <a href="glib_000.html#K25" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="glib_000.html#K26" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E1:188_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="glib_000.html#D39" target="_self">Def39</a></i>;<br/><a NAME="E4:188_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="glib_000.html#M2" target="_self">Subgraph</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:188"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_000.html#T94" target="_self">Th94</a></i>;<br/><b>hence </b><a NAME="E5:188_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="glib_000.html#M3" target="_self">inducedSubgraph</a> of <font color="Maroon">c<sub>2</sub></font>,<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_000.html#E2:188_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="glib_000.html#E3:188_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="glib_000.html#D39" target="_self">Def39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="188_1_1_2"><b>suppose </b></a><a NAME="E1:188_1_1_2"/><i><font color="Green">E79</font></i>: 
(  not <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  not <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:188_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
(  not <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> or  not <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E1:188"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_000.html#D36" target="_self">Def36</a>, <a class="ref" href="glib_000.html#T93" target="_self">Th93</a></i>;<br/><a NAME="E3:188_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R5" target="_self">==</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:188_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="glib_000.html#D39" target="_self">Def39</a></i>;<br/><a NAME="E4:188_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R5" target="_self">==</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:188"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_000.html#T88" target="_self">Th88</a></i>;<br/><a NAME="E5:188_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is    <a href="glib_000.html#M2" target="_self">Subgraph</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T90" target="_self">Th90</a></i>;<br/><b>hence </b><a NAME="E6:188_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="glib_000.html#M3" target="_self">inducedSubgraph</a> of <font color="Maroon">c<sub>2</sub></font>,<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_000.html#E2:188_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="glib_000.html#E4:188_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="glib_000.html#D39" target="_self">Def39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:188"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="glib_000.html#M3" target="_self">inducedSubgraph</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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