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<b>let </b><font color="Maroon">G1</font> be   <a href="glib_000.html#NM2">_Graph</a>, <font color="Maroon">G2</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:67"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2</font>
 ;<br/>

<b>let </b><font color="Maroon">u1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">u2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>;<br/>



<b>assume </b><a NAME="E2:67"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font>
 ;<br/>

<b>let </b><font color="Maroon">H1</font> be    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G1</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u1</font></span><a href="domain_1.html#K6">}</a></span>;<br/><b>let </b><font color="Maroon">H2</font> be    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G2</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u2</font></span><a href="domain_1.html#K6">}</a></span>;<br/>



<b>set </b><font color="Maroon">G1Adj</font> = <font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">G2Adj</font> = <font color="Maroon">G2</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">u2</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E3:67"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">H1</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:67"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G2</font>,<span class="p1">(<span class="default"><font color="Maroon">G2</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">u2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:67"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</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">u2</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="chord.html#T12" target="_self">Th12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1"/>
(  not <font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span> 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">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> or <font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span> 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">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>suppose </b></a><a NAME="E1:67_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span> is  not  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">G1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_1_1"/>
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:67"><i><font color="Green">E56</font></i></a>, <a class="txt" href="chord.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:67_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E4:67_1_1"/>
<font color="Maroon">H2</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E5:67_1_1"/>
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><b>suppose </b></a><a NAME="E1:67_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span> 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">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_1_2"/><i><font color="Green">E69</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">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="chord.html#E4:67"><i><font color="Green">E56</font></i></a>, <a class="txt" href="chord.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:67_1_2"/><i><font color="Green">E70</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">G2</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">u2</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">G2</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">u2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="chord.html#E5:67"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E4:67_1_2"/>
( <font color="Maroon">G1</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G2</font> &amp; <font color="Maroon">G2</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T90">GLIB_000:90</a></i>;<br/><a NAME="E5:67_1_2"/><b>then </b>
( <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">G2</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">G1</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">u1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T79">GLIB_000:79</a></i>;<br/><a NAME="E6:67_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H2</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">H2</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T15" target="_self">Th15</a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E7:67_1_2"/>
( <font color="Maroon">H2</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G2</font> &amp; <font color="Maroon">G2</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T90">GLIB_000:90</a></i>;<br/><a NAME="E8:67_1_2"/><b>then </b>
<font color="Maroon">H2</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T46">GLIB_000:46</a></i>;<br/><b>hence </b><a NAME="E9:67_1_2"/>
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T89">GLIB_000:89</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
