<?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_003.html#V7">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/>

<b>set </b><font color="Maroon">PCS</font> =  <a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  = 
<span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="glib_004.html#T3" target="_self">Th3</a></i>
;<br/><b>hence </b><a NAME="E2:68_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  = 
<span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
<b>by </b><i/>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="glib_004.html#T3" target="_self">Th3</a></i>
;<br/><b>hence </b><a NAME="E4:68_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:68"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>now </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="E1:68_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>set </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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>set </b><font color="Maroon">Next</font> =  <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>;<br/><b>set </b><font color="Maroon">e</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/><a NAME="E2:68_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K13" target="_self">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1"/>
(  <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_1"/>
 <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:68_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_004.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E3:68_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">src</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">tar</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:68_2_1_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:68_2"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E3:68_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="glib_003.html#T41">GLIB_003:41</a></i>;<br/><a NAME="E4:68_2_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></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"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E5:68_2_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E1:68_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E6:68_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E7:68_2_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_000.html#K10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_000.html#K11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E8:68_2_1_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R3">SJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:68_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="glib_004.html#E1:68_2"><i><font color="Green">E35</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or  not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><a NAME="E2:68_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i>, <a class="txt" href="glib_004.html#E2:68_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_004.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E3:68_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T63">GLIB_003:63</a></i>;<br/><a NAME="E4:68_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:68_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_004.html#E1:68_2_1_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K11">.labeledE()</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
<b>by </b><i><a class="txt" href="glib_004.html#E1:68_2_1_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_003.html#T56">GLIB_003:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="glib_004.html#E1:68_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_003.html#T52">GLIB_003:52</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_2_1_1_2_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E5:68_2_1_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="glib_004.html#E2:68_2_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="glib_004.html#E2:68_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_004.html#E4:68_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_004.html#E5:68_2_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:68_2_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:68_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:68_2_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E8:68_2_1_1_2_1_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E2:68_2_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E9:68_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T39">GLIB_000:39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_2_1_1_2_1_1_1_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1_2_1_1_1_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E4:68_2_1_1_2_1_1_1"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div class="add"><a NAME="E2:68_2_1_1_2_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:68_2_1_1_2_1_1_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:68_2_1_1_2_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:68_2_1_1_2_1_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:68_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:68_2_1_1_2_1_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E2:68_2_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:68_2_1_1_2_1_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:68_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E5:68_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="glib_004.html#E8:68_2_1_1_2_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="glib_004.html#E8:68_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_004.html#E5:68_2_1_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_000.html#T34">GLIB_000:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:68_2_1_1_2_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:68_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_2"/><i><font color="Green">E65</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><a NAME="E2:68_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i>, <a class="txt" href="glib_004.html#E2:68_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_004.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E3:68_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T63">GLIB_003:63</a></i>;<br/><a NAME="E4:68_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:68_2_1_1_2"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><i><font color="Green">E68</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K11">.labeledE()</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
<b>by </b><i>, <a class="ref" href="glib_003.html#T56">GLIB_003:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="glib_004.html#E1:68_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_003.html#T52">GLIB_003:52</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_2_1_1_2_1_1_2_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:68_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_004.html#E4:68_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_004.html#E5:68_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:68_2_1_1_2_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:68_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E8:68_2_1_1_2_1_1_2"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E9:68_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T39">GLIB_000:39</a></i>;<br/><a NAME="E10:68_2_1_1_2_1_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E8:68_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_004.html#E7:68_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_000.html#D17">GLIB_000:def 17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_2_1_1_2_1_1_2_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1_2_1_1_2_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div class="add"><a NAME="E2:68_2_1_1_2_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:68_2_1_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:68_2_1_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:68_2_1_1_2_1_1_2_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:68_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:68_2_1_1_2_1_1_2_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:68_2_1_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:68_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E5:68_2_1_1_2"><i><font color="Green">E51</font></i></a>, , , , <a class="ref" href="glib_000.html#T34">GLIB_000:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:68_2_1_1_2_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:68_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" 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> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:68_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:68_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:68"/><b>then </b><i><font color="Green">E75</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:68"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:68"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> )
 ;<br/>


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