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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">S</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>let </b><font color="Maroon" title="c3">H</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">S</font>;<br/><b>let </b><font color="Maroon" title="c4">u</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E55">A1</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E56">A2</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c5">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">H</font>;<br/>

<b>assume </b><a NAME="E3:68"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">Ga</font> be    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><b>let </b><font color="Maroon" title="c7">Ha</font> be    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c3">H</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>



<a NAME="E4:68"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E1:68"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="chord.html#E3:68"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="chord.html#T58" target="_self" title="CHORD:th.58">Th58</a></i>;<br/>
<a NAME="E5:68"/>
<font color="Maroon" title="c6">Ga</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#D5" target="_self" title="CHORD:def.5">Def5</a></i>;<br/>
<a NAME="E6:68"/><b>then </b><i><font color="Green" title="E59">A5</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">Ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">Ga</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="chord.html#E2:68"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E7:68"/><i><font color="Green" title="E60">A6</font></i>: 
<font color="Maroon" title="c7">Ha</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">H</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#D5" target="_self" title="CHORD:def.5">Def5</a></i>;<br/>
<a NAME="E8:68"/><b>then </b><i><font color="Green" title="E61">A7</font></i>: 
(  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c7">Ha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c7">Ha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="chord.html#E4:68"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="chord.html#E2:68"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<b>thus </b><a NAME="E9:68"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">Ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c7">Ha</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:68"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="chord.html#E7:68"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="chord.html#E4:68"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a><br/>

<b>thus </b><a NAME="E10:68"/><i><font color="Green" title="E62">A8</font></i>: 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">Ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c7">Ha</font>
 <b>by </b><i><a class="txt" href="chord.html#E6:68"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="chord.html#E8:68"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="chord.html#E4:68"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="chord.html#E1:68"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="chord.html#T31" target="_self" title="CHORD:th.31">Th31</a></i>;<br/>

<a NAME="E11:68"/><i><font color="Green" title="E63">A9</font></i>: 
(  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">Ga</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="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">Ga</font></span>)</span> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">Ga</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="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">Ga</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="glib_000.html#T48" title="GLIB_000:th.48">GLIB_000:48</a></i>;<br/>
<a NAME="E12:68"/>
<font color="Maroon" title="c7">Ha</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">H</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#D5" target="_self" title="CHORD:def.5">Def5</a>, <a class="txt" href="chord.html#E4:68"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/>
<a NAME="E13:68"/><b>then </b>
<font color="Maroon" title="c7">Ha</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E1:68"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="chord.html#E2:68"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="chord.html#T29" target="_self" title="CHORD:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E14:68"/>
(  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c7">Ha</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">Ga</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c7">Ha</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">Ga</font> )
 <b>by </b><i><a class="txt" href="chord.html#E11:68"><i><font color="Green" title="E63">A9</font></i></a>, <a class="txt" href="chord.html#E10:68"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="glib_000.html#T48" title="GLIB_000:th.48">GLIB_000:48</a></i>;<br/>


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