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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>set </b><font color="Maroon">CS</font> =  <a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> ;<br/>
<a NAME="E1:219"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a>  is <a href="lexbfs.html#V21" target="_self">with_property_L3</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="219_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span>, <font color="Maroon">c</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:219_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="chord.html#R1">are_adjacent</a>  )
 ;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span><b> such that </b><br/><a NAME="E3:219_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="chord.html#R1">are_adjacent</a>  &amp; ( for <font color="Olive">e</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span>  st <font color="Olive">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Olive">e</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">e</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> ) )
 <b>by </b><i><a class="txt" href="lexbfs.html#E19"><i><font color="Green">Lemma127</font></i></a>, <a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">d</font> = <font color="Maroon">d</font>;<br/><b>thus </b><a NAME="E4:219_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:219_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><a NAME="E6:219_1"/>
<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:219_1"/>
not <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:219"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K19" target="_self">.Result()</a>  is <a href="lexbfs.html#V22" target="_self">with_property_T</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a></i>;<br/>


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