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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">LexBFS:Labeling</a> of <font color="Maroon" title="c1">G</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></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> holds <br/> not  <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Olive" title="b1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">LexBFS:Labeling</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></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#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:72"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">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/>
<span class="kw">set </span><font color="Maroon" title="c4">V2G</font> = <font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">VLG</font> = <font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">w</font> =  <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>, <font color="Maroon" title="c8">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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>, <font color="Maroon" title="c9">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:72"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">and </span><br/><a NAME="E4:72"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:72"/>
 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#NK1" title="NUMBERS:NK.1">NAT</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font> holds <br/>(<font color="Olive" title="b1">x</font>,1) <a href="uproots.html#K1" title="UPROOTS:func.1">-bag</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">and </span><br/><a NAME="E6:72"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">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#NK1" title="NUMBERS:NK.1">NAT</a> st <br/>( <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">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#K1" title="UPROOTS:func.1">-bag</a>  )
 <span class="kw">and </span><br/><a NAME="E7:72"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
 <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> (<font color="Maroon" title="c8">B</font>,<span class="p4">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>)</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="lexbfs.html#D12" target="_self" onmouseover="rs('lexbfs/D12')" onmouseout="rh()">Def11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">mw</font> =  <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> (<font color="Maroon" title="c8">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>);<br/>
<a NAME="E8:72"/>
 <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> (<font color="Maroon" title="c8">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lexbfs.html#D5" target="_self" onmouseover="rs('lexbfs/D5')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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#NK1" title="NUMBERS:NK.1">NAT</a><span class="kw"> such that </span><br/><a NAME="E10:72"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">and </span><br/><a NAME="E11:72"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
 <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> (<font color="Maroon" title="c8">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>) <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> (<font color="Maroon" title="c11">y</font>,1) <a href="uproots.html#K1" title="UPROOTS:func.1">-bag</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72"><span class="lab"><font color="Green" title="E52">A4</font></span></a></span>;<br/>
<a NAME="E12:72"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> (<font color="Maroon" title="c8">B</font>,<span class="p2">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="uproots.html#T8" onmouseover="rs('uproots/T8')" onmouseout="rh()">UPROOTS:8</a></span>;<br/>
<a NAME="E13:72"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> (<font color="Maroon" title="c8">B</font>,<span class="p4">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>)</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="funct_1.html#T72" onmouseover="rs('funct_1/T72')" onmouseout="rh()">FUNCT_1:72</a></span>;<br/>
<a NAME="E14:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A8</font></span>: 
 <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E15:72"/>
 <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E16:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A9</font></span>: 
 not  <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="tarski.html#R2" title="TARSKI: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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E17:72"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:72"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E16:72"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
