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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">A</font> be  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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>, <font color="Maroon">B</font> be  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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<b>let </b><font color="Maroon">H1</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">H2</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">B</font>;<br/>



<a NAME="E2:33"/><i><font color="Green">E33</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green">E53</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green">E56</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green">E59</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E66</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H1</font>
 
<b>by </b><i>, , <a class="txt" href="chord.html#E2:33"><i><font color="Green">E33</font></i></a>, <a class="ref" href="glib_000.html#T39">GLIB_000:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">H1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">H1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">H2</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">H1</font>
 
<b>by </b><i>, <a class="txt" href="chord.html#E1:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="chord.html#T2" target="_self">Th2</a>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/>
<a NAME="E9:33"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">H1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T79">GLIB_000:79</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:33_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">H1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><a NAME="E3:33_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">H1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><a NAME="E4:33_2"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T34">GLIB_000:34</a></i>;<br/><b>hence </b><a NAME="E5:33_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="chord.html#T3" target="_self">Th3</a>, , , <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#T34">GLIB_000:34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:33"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:33"/><b>then </b>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E13:33"/>
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H1</font>,<font color="Maroon">B</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E1:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="chord.html#T2" target="_self">Th2</a>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>


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