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

<b>set </b><font color="Maroon" title="c2">PCS</font> =  <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c1">G</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>;<br/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:Init</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self" title="GLIB_004:def.15">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<a NAME="E2:60_1_1"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E39">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:60_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">Gn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font> as  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V11" title="GLIB_003:attr.11">real-WEV</a> <a href="glib_003.html#NM7" title="GLIB_003:NM.7">WEVGraph</a> ;<br/><b>set </b><font color="Maroon" title="c5">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c6">Next</font> =  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font>;<br/><b>set </b><font color="Maroon" title="c7">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span>;<br/><a NAME="E3:60_1_1_2"/><i><font color="Green" title="E40">A3</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c4">Gn</font>
 <b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self" title="GLIB_004:def.15">Def15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_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="E1:60_1_1_2_1_1"/>
(  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or (  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">Gn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Gn</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  ) or (  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">Gn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Gn</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:60_1_1_2_1_1_1"/>
 <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_004.html#E3:60_1_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self" title="GLIB_004:def.14">Def14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:60_1_1_2_1_1_2"/>
(  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">Gn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Gn</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:60_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Gn</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">Gn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="glib_004.html#E3:60_1_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self" title="GLIB_004:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:60_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:60_1_1_2_1_1_3"/>
(  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">Gn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Gn</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:60_1_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Gn</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">Gn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c4">Gn</font></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="glib_004.html#E3:60_1_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self" title="GLIB_004:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:60_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:60_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:60_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_004.html#E2:60_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="glib_004.html#E3:60_1_1"><i><font color="Green" title="E39">A2</font></i></a>);<br/></i>
<a NAME="E5:60_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:60_1_1_3"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:60_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E4:60_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:60_1_1"/>
 <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font> is <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D60" title="GLIB_000:def.60">GLIB_000:def 60</a></i>;<br/>


</div>
