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

<b>let </b><font color="Maroon">GS</font> be   <a href="lexbfs.html#NM10">VLabelNumberingSeq</a>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:129"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>and </b><br/><a NAME="E2:129"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>and </b><br/><a NAME="E3:129"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font>
 ;<br/>

<b>set </b><font color="Maroon">CSI</font> = <font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">GN</font> = <font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> ;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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">CSJ</font> = <font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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>;<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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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></span>)</span>;<br/>
<b>set </b><font color="Maroon">CSK</font> = <font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">VLK</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E4:129"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:129"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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="E6:129"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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#R1">&lt;=</a> <font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:129"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E8:129"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E9:129"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> 
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<b>assume </b><a NAME="E10:129"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/>

<a NAME="E11:129"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">GS</font> <a href="lexbfs.html#K10" target="_self">.PickedAt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E12:129"/>
<span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D1" target="_self">Def1</a>, , , <a class="ref" href="xreal_1.html#T17">XREAL_1:17</a></i>;<br/>
<a NAME="E13:129"/><b>then </b>
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E2"><i><font color="Green">Lemma52</font></i></a></i>;<br/>
<a NAME="E14:129"/><b>then </b>
 <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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" 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></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<b>hence </b><a NAME="E15:129"/>
contradiction
 <b>by </b><i><a class="ref" href="lexbfs.html#D2" target="_self">Def2</a>, , , </i>;<br/>


</div>
