<?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>set </b><font color="Maroon">CS</font> =  <a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font>;<br/>
<a NAME="E1:159"/>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:159"/><b>then </b><i><font color="Green">E34</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <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="lexbfs.html#K9" target="_self">.</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="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E12"><i><font color="Green">Lemma78</font></i></a></i>;<br/>
<a NAME="E3:159"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <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="Olive">n</font> <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="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="159_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:159_1"/><i><font color="Green">E36</font></i>: 
<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">n</font> <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="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:159_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 ;<br/>

<b>set </b><font color="Maroon">VLN</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">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">VN1</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">w</font> =  <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">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">j</font> =  <a href="card_1.html#K4">card</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> <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">n</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E3:159_1"/><i><font color="Green">E39</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">n</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">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:159_1"/>
 <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">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D5" target="_self">Def5</a>, <a class="ref" href="lexbfs.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:159_1"/><b>then </b><i><font color="Green">E40</font></i>: 
not  <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">n</font></span>)</span> <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">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E6:159_1"/><i><font color="Green">E42</font></i>: 
 <a href="card_1.html#K4">card</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> <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">n</font></span>)</span></span>)</span></span>)</span> <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="lexbfs.html#D5" target="_self">Def5</a>, <a class="ref" href="lexbfs.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E7:159_1"/><i><font color="Green">E45</font></i>: 
 <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</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">n</font></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><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">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">wf</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</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">n</font></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</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="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<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">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E8:159_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</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">n</font></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><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">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</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">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</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">n</font></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><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">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><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">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E9:159_1"/><b>then </b><i><font color="Green">E46</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> <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> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="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">n</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</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">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E10:159_1"/>
 <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">n</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</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">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:159_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:159"/>
<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>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i>, <a class="ref" href="glib_000.html#D57">GLIB_000:def 57</a></i>;<br/>


</div>
