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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>, <font color="Maroon">H</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G1</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">H1</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G1</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">H2</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H1</font>,<font color="Maroon">C</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">C</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<a NAME="E4:29"/><i><font color="Green">E39</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<a NAME="E5:29"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
( ( <font color="Maroon">A</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">B</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> ) or ( <font color="Maroon">A</font> 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">G</font></span>)</span> &amp; <font color="Maroon">B</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> ) or ( <font color="Maroon">A</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">B</font> 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">G</font></span>)</span> ) or ( <font color="Maroon">A</font> 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">G</font></span>)</span> &amp; <font color="Maroon">B</font> 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">G</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">A</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">B</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:29_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">G2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="ref" href="chord.html#T29">CHORD:29</a></i>;<br/><a NAME="E3:29_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="chord.html#T29">CHORD:29</a></i>;<br/><a NAME="E4:29_1_1"/>
<font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><a NAME="E5:29_1_1"/><b>then </b>
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T98">GLIB_000:98</a></i>;<br/><b>hence </b><a NAME="E6:29_1_1"/>
<font color="Maroon">G2</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T96">GLIB_000:96</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">A</font> 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">G</font></span>)</span> &amp; <font color="Maroon">B</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:29_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:29_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><a NAME="E4:29_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">G2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T98">GLIB_000:98</a></i>;<br/><a NAME="E5:29_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="chord.html#T29">CHORD:29</a></i>;<br/><a NAME="E6:29_1_2"/>
<font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><a NAME="E7:29_1_2"/><b>then </b>
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T98">GLIB_000:98</a></i>;<br/><b>hence </b><a NAME="E8:29_1_2"/>
<font color="Maroon">G2</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T96">GLIB_000:96</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_3"><b>suppose </b></a><a NAME="E1:29_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">A</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">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">B</font> 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">G</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:29_1_3"/><b>then </b>
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:29_1_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E4:29_1_3"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><a NAME="E5:29_1_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T98">GLIB_000:98</a></i>;<br/><a NAME="E6:29_1_3"/>
<font color="Maroon">G2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="ref" href="chord.html#T29">CHORD:29</a></i>;<br/><b>hence </b><a NAME="E7:29_1_3"/>
<font color="Maroon">G2</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T96">GLIB_000:96</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_4"><b>suppose </b></a><a NAME="E1:29_1_4"/>
( <font color="Maroon">A</font> 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">G</font></span>)</span> &amp; <font color="Maroon">B</font> 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">G</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:29_1_4"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> &amp; <font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:29_1_4"/><b>then </b>
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E4:29_1_4"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E5:29_1_4"/><b>then </b>
<font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><a NAME="E6:29_1_4"/><b>then </b>
<font color="Maroon">H2</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G1</font>,<font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T98">GLIB_000:98</a></i>;<br/><b>hence </b><a NAME="E7:29_1_4"/>
<font color="Maroon">G2</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T96">GLIB_000:96</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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