<?xml version="1.0"?>
<div class="add">

<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>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><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#K9" target="_self">.</a> <font color="Maroon">i</font></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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:175"/><i><font color="Green">E34</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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:175"/><i><font color="Green">E36</font></i>: 
<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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:175"/><i><font color="Green">E38</font></i>: 
<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#K9" target="_self">.</a> <font color="Maroon">i</font></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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:175"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">CSI</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#K9" target="_self">.</a> <font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">CSJ</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#K9" target="_self">.</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">VLI</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">VLJ</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">V2I</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">V2J</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span>;<br/>
<a NAME="E5:175"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:175"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:175"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <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="glib_000.html#K40">.Lifespan()</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:175"/>
<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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 
<b>by </b><i>, <a class="txt" href="lexbfs.html#E15:175"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E9:175"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E10:175"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a>, , </i>;<br/>
<a NAME="E11:175"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<a NAME="E12:175"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E13:175"/>
<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#K10" target="_self">.PickedAt</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:175"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:175"/><i><font color="Green">E50</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> <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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:175"/>
( <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#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="glib_000.html#R5">==</a> <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#K9" target="_self">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <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#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E17:175"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:175_1"/>
<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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:175_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E3:175_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#K9" target="_self">.</a> <font color="Maroon">i</font></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#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E4:175_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:175"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">a</font></span>)</span>,1 <a href="uproots.html#K2">-bag</a>  <a href="termord.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">b</font></span>)</span>,1 <a href="uproots.html#K2">-bag</a> , <a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, , , </i>;<br/>


</div>
