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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="lexbfs.html#V5" target="_self">natural-v2labeled</a> <a href="lexbfs.html#NM3">VVGraph</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:193"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">x</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> <font color="Maroon">G</font></span>)</span>
 <b>and </b><br/><a NAME="E2:193"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E3:193"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/>

<b>set </b><font color="Maroon">w</font> =  <a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">VLG</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">V2G</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">VG</font> =  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>;<br/>
<b>consider </b><font color="Maroon">S</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V5">natural-membered</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:193"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E6:193"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:193"/><i><font color="Green">E42</font></i>: 
 <a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D6" target="_self">Def6</a>, , </i>;<br/>
<b>set </b><font color="Maroon">y</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font>;<br/>
<a NAME="E8:193"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E9:193"/><b>then </b><i><font color="Green">E45</font></i>: 
not <font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/>
<a NAME="E10:193"/><i><font color="Green">E46</font></i>: 
(  <a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E11:193"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E12:193"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:193"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:193_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1_1"><b>suppose </b></a><a NAME="E1:193_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:193_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:193_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:193_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E5:193_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:193_1_1"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="193_1_2"><b>suppose </b></a><a NAME="E1:193_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:193_1_2"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a>, , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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