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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> implies  not  <a href="lexbfs.html#K13" title="LEXBFS:func.13">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:148"/><i><font color="Green" title="E73">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E2:148"/><i><font color="Green" title="E74">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="lexbfs.html#K13" title="LEXBFS:func.13">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c2">w</font> =  <a href="lexbfs.html#K13" title="LEXBFS:func.13">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">VLG</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c4">V2G</font> =  <a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c5">VG</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>assume </b><a NAME="E3:148"/><i><font color="Green" title="E75">A3</font></i>: 
 <a href="lexbfs.html#K13" title="LEXBFS:func.13">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c6">S</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>, <font color="Maroon" title="c7">B</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>, <font color="Maroon" title="c8">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:148"/><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font>
 <b>and </b><br/><a NAME="E6:148"/><i><font color="Green" title="E77">A5</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_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"><span class="p2">(<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 href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:148"/>
for <font color="Olive" title="b1">x</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> holds <br/><font color="Olive" title="b1">x</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <b>and </b><br/><a NAME="E8:148"/><i><font color="Green" title="E78">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> holds <br/> ex <font color="Olive" title="b2">y</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  )
 <b>and </b><br/><a NAME="E9:148"/><i><font color="Green" title="E79">A7</font></i>: 
 <a href="lexbfs.html#K13" title="LEXBFS:func.13">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c7">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:148"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="lexbfs.html#E2:148"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="lexbfs.html#D29" target="_self" title="LEXBFS:def.29">Def29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">mw</font> =  <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c7">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>;<br/>
<a NAME="E10:148"/>
 <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c7">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D5" target="_self" title="LEXBFS:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">y</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:148"/><i><font color="Green" title="E80">A8</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> &amp;  <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c7">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c10">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  )
 <b>by </b><i><a class="txt" href="lexbfs.html#E8:148"><i><font color="Green" title="E78">A6</font></i></a></i>;<br/>
<a NAME="E13:148"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c7">B</font>,<span class="p2">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E12:148"><i><font color="Green" title="E80">A8</font></i></a>, <a class="ref" href="uproots.html#T10" title="UPROOTS:th.10">UPROOTS:10</a></i>;<br/>
<a NAME="E14:148"/><b>then </b>
not <font color="Maroon" title="c8">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c7">B</font>,<span class="p4">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E5:148"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="lexbfs.html#E12:148"><i><font color="Green" title="E80">A8</font></i></a>, <a class="ref" href="funct_1.html#T142" title="FUNCT_1:th.142">FUNCT_1:142</a></i>;<br/>
<a NAME="E15:148"/><b>then </b><i><font color="Green" title="E81">A9</font></i>: 
 <a href="lexbfs.html#K13" title="LEXBFS:func.13">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">F</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E9:148"><i><font color="Green" title="E79">A7</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E16:148"/><i><font color="Green" title="E82">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<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 href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E6:148"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E17:148"/>
not  <a href="lexbfs.html#K13" title="LEXBFS:func.13">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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 href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E3:148"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E18:148"/>
contradiction
 <b>by </b><i><a class="txt" href="lexbfs.html#E15:148"><i><font color="Green" title="E81">A9</font></i></a>, <a class="txt" href="lexbfs.html#E16:148"><i><font color="Green" title="E82">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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