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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#V6" target="_self">finite-v2labeled</a> <a href="lexbfs.html#V7" target="_self">natsubset-v2labeled</a> <a href="lexbfs.html#NM3">VVGraph</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:148"/><i><font color="Green">E34</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="E2:148"/><i><font color="Green">E36</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#K13" target="_self">LexBFS: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>assume </b><a NAME="E3:148"/><i><font color="Green">E38</font></i>: 
 <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <font color="Maroon">G</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>
 ;<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="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="numbers.html#K5">NAT</a> </span>)</span>, <font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:148"/><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="E6:148"/><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="E7:148"/>
for <font color="Olive">x</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">x</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:148"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> ex <font color="Olive">y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>,1 <a href="uproots.html#K2">-bag</a>  )
 <b>and </b><br/><a NAME="E9:148"/><i><font color="Green">E45</font></i>: 
 <a href="lexbfs.html#K13" target="_self">LexBFS: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="polynom1.html#K11">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T7" target="_self">Th7</a>, <a class="ref" href="lexbfs.html#T6" target="_self">Th6</a>, <a class="ref" href="lexbfs.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">mw</font> =  <a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span>;<br/>
<a NAME="E10:148"/>
 <a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:148"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  <a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">y</font>,1 <a href="uproots.html#K2">-bag</a>  )
 <b>by </b><i><a class="txt" href="lexbfs.html#E12"><i><font color="Green">Lemma78</font></i></a></i>;<br/>
<a NAME="E13:148"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lexbfs.html#T8" target="_self">Th8</a>, <a class="ref" href="uproots.html#T10">UPROOTS:10</a></i>;<br/>
<a NAME="E14:148"/><i><font color="Green">E48</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="polynom1.html#K11">support</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p4">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></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="lexbfs.html#T8" target="_self">Th8</a>, , <a class="ref" href="lexbfs.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/>
<a NAME="E15:148"/><i><font color="Green">E49</font></i>: 
 <a href="lexbfs.html#K13" target="_self">LexBFS: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>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="txt" href="lexbfs.html#E13"><i><font color="Green">Lemma79</font></i></a>, </i>;<br/>
<a NAME="E16:148"/><i><font color="Green">E50</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="lexbfs.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E17:148"/>
not  <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <font color="Maroon">G</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/>
<b>hence </b><a NAME="E18:148"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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