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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">H1</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c5">H2</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">B</font>;<br/>



<a NAME="E2:33"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c5">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E5:33"/>
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">H1</font>
 
<b>by </b><i><a class="txt" href="chord.html#E1:33"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="chord.html#E4:33"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="glib_000.html#T39" title="GLIB_000:th.39">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" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">H2</font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font>
<b>by </b><i><a class="txt" href="chord.html#E1:33_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font>
<b>by </b><i><a class="txt" href="chord.html#E6:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="chord.html#E1:33_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font>
<b>by </b><i><a class="txt" href="chord.html#E1:33_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font>
<b>by </b><i><a class="txt" href="chord.html#E6:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="chord.html#E1:33_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:33"/><b>then </b><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c5">H2</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c4">H1</font>
 
<b>by </b><i><a class="txt" href="chord.html#E1:33"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="chord.html#E2:33"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="chord.html#E3:33"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="chord.html#E6:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>;<br/>
<a NAME="E9:33"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c4">H1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">H2</font>
 
<b>by </b><i><a class="txt" href="chord.html#E4:33"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="glib_000.html#T79" title="GLIB_000:th.79">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" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_2"/><i><font color="Green" title="E36">A9</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">H2</font>
 ;<br/><a NAME="E2:33_2"/><i><font color="Green" title="E37">A10</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="chord.html#E1:33_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>;<br/><a NAME="E3:33_2"/><i><font color="Green" title="E38">A11</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="chord.html#E1:33_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>;<br/><a NAME="E4:33_2"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="chord.html#E4:33"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="chord.html#E1:33_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="glib_000.html#T34" title="GLIB_000:th.34">GLIB_000:34</a></i>;<br/><b>hence </b><a NAME="E5:33_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="chord.html#E1:33_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="chord.html#E2:33_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="chord.html#E3:33_2"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="glib_000.html#T34" title="GLIB_000:th.34">GLIB_000:34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:33"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">H2</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E9:33"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>
<a NAME="E12:33"/><b>then </b>
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E13:33"/>
<font color="Maroon" title="c5">H2</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:33"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="chord.html#E2:33"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="chord.html#E3:33"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="chord.html#E8:33"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>


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