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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">S</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>let </b><font color="Maroon">H</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>;<br/>

<b>assume </b><a NAME="E3:68"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/>

<b>let </b><font color="Maroon">Ga</font> be    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span>;<br/><b>let </b><font color="Maroon">Ha</font> be    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">H</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>;<br/>



<a NAME="E4:68"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:68"/>
<font color="Maroon">Ga</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:68"/><b>then </b><i><font color="Green">E59</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ga</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Ga</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="chord.html#E4:68"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E7:68"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">Ha</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H</font>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:68"/><b>then </b><i><font color="Green">E67</font></i>: 
(  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Ha</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp;  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ha</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="txt" href="chord.html#E4:68"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<b>thus </b><a NAME="E9:68"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ga</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Ha</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:68"><i><font color="Green">E59</font></i></a>, <a class="ref" href="chord.html#T15" target="_self">Th15</a>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a><br/>

<b>thus </b><a NAME="E10:68"/><i><font color="Green">E68</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Ga</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Ha</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:68"><i><font color="Green">E59</font></i></a>, , , , </i>;<br/>

<a NAME="E11:68"/><i><font color="Green">E69</font></i>: 
(  <a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">Ga</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">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Ga</font></span>)</span> &amp;  <a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">Ga</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">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Ga</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="glib_000.html#T48">GLIB_000:48</a></i>;<br/>
<a NAME="E12:68"/>
<font color="Maroon">Ha</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:68"/><b>then </b>
<font color="Maroon">Ha</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="chord.html#E4:68"><i><font color="Green">E56</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E14:68"/>
(  <a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">Ha</font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">Ga</font> &amp;  <a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">Ha</font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">Ga</font> )
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T48">GLIB_000:48</a></i>;<br/>


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