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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <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/> for <font color="Olive" title="b2">H</font> being  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">S</font> holds  <font color="Olive" title="b2">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <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> holds  <font color="Olive" title="b1">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1"/>
( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_1"/>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/><div class="add"><a NAME="E2:73_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">H</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><span class="kw">hence </span><a NAME="E3:73_1_1"/>
<font color="Maroon" title="c3">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T62" target="_self" title="CHORD:th.62">Th62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_2"/>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/><div class="add"><a NAME="E2:73_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A1</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">u</font>, <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font> implies <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_2_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">v2</font> = <font color="Maroon" title="c5">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_2"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">u2</font> = <font color="Maroon" title="c4">u</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_2"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E4:73_1_2_1"/>
<font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c6">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_2_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:73_1_2_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E7:73_1_2_1"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_2"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E6:73_1_2_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E8:73_1_2_1"/>
<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:73_1_2"/>
<font color="Maroon" title="c3">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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